THE ALGEBRA OF SECONDARY COHOMOLOGY OPERATIONS AND ITS DUAL

HANS JOACHIM BAUES AND MAMUKA JIBLADZE

We describe a modification ℬ^{}of the algebra ℬ of secondary cohomology operations in [1] which is suitable for dualization. The resulting dual object ℬ_{}will be used to give an alternative description of the E_{3}-term of the Adams spectral sequence, which complements the algorithm described in [1, 2]. All triple Massey products in the Steenrod algebra can be deduced from ℬ^{}or ℬ_{}.

In Part I we first recall the notions of pair modules and pair algebras from [2] and give the corresponding dual notions. Next we define the concept of M-algebras and N-coalgebras, where M is a folding system and N an unfolding system. An M-algebra is a variation on the notion of a [p]-algebra from [1]. We show that the algebra ℬ of secondary cohomology operations in [1] gives rise to a comonoid ℬ^{}in the monoidal category of M-algebras, and we describe the dual object ℬ_{}, which is a monoid in the monoidal category of N-coalgebras.

In Part II we study the algebraic objects ℬ^{}and ℬ_{}in terms of generators. This way we obtain explicit descriptions which can be used for computations. In particular we characterize algebraically multiplication maps A_{ϕ} and comultiplication maps A^{ψ} which determine ℬ^{}and ℬ_{}completely, see sections 15, 16, 17. For the dual object ℬ_{}the inclusion of polynomial algebras 𝒜_{∗} ⊂ ℱ_{∗} is crucial. Here 𝒜_{∗} is the Milnor dual of the Steenrod algebra and ℱ_{∗} is the dual of a free associative algebra.

Part I. Hopf pair algebras and Hopf pair coalgebras representing the algebra of secondary cohomology operations

1. Pair modules and pair algebras

We here recall from [1, 2] the notion of pair modules and pair algebras.

Let k be a field (usually it will be actually a prime field = _{p} = /p for some prime p) and let Mod be the category of finite dimensional k-modules (i. e. k-vector spaces) and k-linear maps. This is a symmetric monoidal category via the tensor product A⊗B over k of k-modules A and B. A pair module is a homomorphism

(1.1)  X = (X_{1} ⟶^{∂} X_{0})

in Mod. We write π_{0}(X) = coker∂ and π_{1}(X) = ker∂. A morphism f : X → Y of pair modules is a commutative diagram

\xymatrix{X_{1} \ar[r]^{f_{1}} \ar[d]_{∂} & Y_{1} \ar[d]^{∂} \\ X_{0} \ar[r]^{f_{0}} & Y_{0}.}

Evidently pair modules with these morphisms form a category Mod_{∗} and one has functors

π_{0}, π_{1} : Mod_{∗} → Mod.

A morphism of pair modules is called a weak equivalence if it induces isomorphisms on π_{0} and π_{1}.

Clearly a pair module is the same as a chain complex concentrated in degrees 0 and 1. For two pair modules X and Y the tensor product of the complexes corresponding to them is concentrated in degrees in 0, 1 and 2 and is given by

(1.2)  X_{1}⊗Y_{1} ⟶^{∂_{1}} X_{1}⊗Y_{0} ⊕ X_{0}⊗Y_{1} ⟶^{∂_{0}} X_{0}⊗Y_{0}

with ∂_{0} = (∂ ⊗ 1, 1 ⊗ ∂) and ∂_{1} = \binom{−1⊗∂}{∂⊗1}. Truncating this chain complex we get the pair module

X ¯⊗Y = ((X ¯⊗Y)_{1} = coker(∂_{1}) ⟶^{∂} X_{0} ⊗ Y_{0} = (X ¯⊗Y)_{0})

with ∂ induced by ∂_{0}.

(1.3) Remark. Note that the full embedding of the category of pair modules into the category of chain complexes induced by the above identification has a left adjoint Tr_{∗} given by truncation: for a chain complex

C_{∗} = (... → C_{2} ⟶^{∂_{1}} C_{1} ⟶^{∂_{0}} C_{0} ⟶^{∂_{−1}} C_{−1} → ...) ,

one has

Tr_{∗}(C_{∗}) = (coker(∂_{1}) ⟶^{\underline{∂}_{0}} C_{0}),

with \underline{∂}_{0} induced by ∂_{0}. Then clearly one has

X ¯⊗Y = Tr_{∗}(X ⊗ Y).

Using the fact that Tr_{∗} is a reflection onto a full subcategory, one easily checks that the category Mod_{∗} together with the tensor product ¯⊗ and unit k_{∗} = (0 → k) is a symmetric monoidal category, and Tr_{∗} is a monoidal functor.

Clearly one has π_{0}(X ¯⊗Y) π_{0}(X) ⊗ π_{0}(Y). We will also need the following

(1.4) Lemma. For any pair modules X, Y there is a natural isomorphism

π_{1}(X ¯⊗Y) π_{1}(X)⊗π_{0}(Y) ⊕ π_{0}(X)⊗π_{1}(Y).

Proof. By the Künneth formula, π_{1}(X)⊗π_{0}(Y) ⊕ π_{0}(X)⊗π_{1}(Y) is isomorphic to the middle homology of the complex (1.2) above. But it is clear that for any homomorphisms ∂_{1} : P → Q, ∂_{0} : Q → R with ∂_{0}∂_{1} = 0 the homology ker∂_{0}/ im∂_{1} is isomorphic to the kernel of the induced map coker(∂_{1}) → R. In our case this kernel is precisely π_{1}(X ¯⊗Y).

We next consider the category Mod^{·}of graded modules, i. e. graded objects in Mod (graded k-vector spaces A^{·}= (A^{n})_{n∈} with upper indices, which in each degree have finite dimension). For graded modules A^{·}, B^{·}we define their graded tensor product A^{·}⊗ B^{·}in the usual way, i. e. by

(A^{·}⊗ B^{·})^{n} = ⨁_{i+j=n} A^{i} ⊗ B^{j}.

This tensor product has an interchange

(1.5)  T_{A^{·},B^{·}}: A^{·}⊗ B^{·} ⟶^{}B^{·}⊗ A^{·}

given on homogeneous elements by T_{A^{·},B^{·}}(a ⊗ b) = (−1)^{deg(a) deg(b)}b ⊗ a.

A graded pair module is a graded object of Mod_{∗}, i. e. a sequence X^{n} = (∂^{n} : X_{1}^{n} → X_{0}^{n}) with n ∈ of pair modules. We can also identify such a graded pair module X^{·}with the underlying morphism ∂ of degree 0 between graded modules

X^{·}= (X_{1}^{·} ⟶^{∂^{·}} X_{0}^{·}).

Now the tensor product X^{·}¯⊗Y^{·}of graded pair modules X^{·}, Y^{·}is defined by

(1.6)  (X^{·}¯⊗Y^{·})^{n} = ⨁_{i+j=n} X^{i} ¯⊗Y^{j}.

This defines a monoidal structure on the category Mod_{∗}^{·} of graded pair modules, with morphisms maps of degree 0. Again ¯⊗ is symmetric.

For two morphisms f, g : X^{·}→ Y^{·}between graded pair modules, a homotopy H : f ⇒ g is a morphism H : X_{0}^{·} → Y_{1}^{·} of degree 0 as in the diagram

(1.7)  [figure: X_{1}^{·}; f_{1}; g_{1}; ∂; Y_{1}^{·}; ∂; X_{0}^{·}; f_{0}; g_{0}; H; Y_{0}^{·},]

satisfying f_{0} − g_{0} = ∂H and f_{1} − g_{1} = H∂.

A pair algebra B^{·}is a monoid in the monoidal category of graded pair modules, with multiplication

μ : B^{·}¯⊗B^{·}→ B^{·}.

We assume that B^{·}is concentrated in nonnegative degrees, that is B^{n} = 0 for n < 0.

More explicitly pair algebras can be described as follows.

(1.8) Definition. A pair algebra B^{·}is a graded pair module, i. e. an object

∂^{·}: B_{1}^{·} → B_{0}^{·}

in Mod_{∗}^{·} with B_{1}^{n} = B_{0}^{n} = 0 for n < 0 such that B_{0}^{·} is a graded algebra in Mod^{·}, B_{1}^{·} is a graded B_{0}^{·}-B_{0}^{·}- bimodule, and ∂^{·}is a bimodule homomorphism. Moreover for x, y ∈ B_{1}^{·} the equality

(1.9)  ∂(x)y = x∂(y)

holds in B_{1}^{·}.

It is easy to see that a graded pair algebra B^{·}yields an exact sequence of graded B_{0}^{·}-B_{0}^{·}-bimodules

(1.10)  0 → π_{1}B^{·}→ B_{1}^{·} ⟶^{∂} B_{0}^{·} → π_{0}B^{·}→ 0

where in fact π_{0}B^{·}is a graded k-algebra, π_{1}B^{·}is a graded π_{0}B^{·}-π_{0}B^{·}-bimodule, and B_{0}^{·} → π_{0}B^{·}is a homomorphism of graded k-algebras.

The tensor product of pair algebras has a natural pair algebra structure, as it happens in any symmetric monoidal category.

We are mainly interested in two examples of pair algebras defined below in sections 5 and 6 respectively: the -relation pair algebra ℛ of the Steenrod algebra 𝒜 and the pair algebra ℬ of secondary cohomology operations deduced from [1, 5.5.2].

By the work of Milnor [7] it is well known that the dual of the Steenrod algebra 𝒜 is a polynomial algebra and this fact yields important algebraic properties of 𝒜. For this reason we also consider the dual of the -relation pair algebra ℛ of 𝒜 and the dual of the pair algebra ℬ of secondary cohomology operations. The duality functor D is studied in the next section.

2. Pair comodules and pair coalgebras

This section is exactly dual to the previous one. There is a contravariant self-equivalence of categories

D = Hom_{k}( , k) : Mod^{op} → Mod

which carries a vector space V in Mod to its dual

DV = Hom_{k}(V, k).

We also denote the dual of V by V_{∗} = DV, for example, the dual of the Steenrod algebra 𝒜 is 𝒜_{∗} = D(𝒜). We can apply the functor Hom_{k}( , k) to dualize straightforwardly all notions of section 1. Explicitly, one gets:

A pair comodule is a homomorphism

(2.1)  X = (X^{1} ⟵^{d} X^{0})

in Mod. We write π^{0}(X) = kerd and π^{1}(X) = cokerd. The dual of a pair module X is a pair comodule

DX= Hom_{k}(X, k)
= (D∂ : DX_{0} → DX_{1})

with (DX)^{i} = D(X_{i}). A morphism f : X → Y of pair comodules is a commutative diagram

\xymatrix{X^{1} \ar[r]^{f^{1}} & Y^{1} \\ X^{0} \ar[u]^{d} \ar[r]^{f_{0}} & Y^{0}. \ar[u]^{d}}

Evidently pair comodules with these morphisms form a category Mod^{∗} and one has functors

π^{0}, π^{1} : Mod^{∗} → Mod.

which are compatible with the duality functor D, that is, for any pair module X one has

π_{i}(DX) = D(π_{i}X) for i = 0, 1.

A morphism of pair comodules is called a weak equivalence if it induces isomorphisms on π^{0} and π^{1}.

Clearly a pair comodule is the same as a cochain complex concentrated in degrees 0 and 1. For two pair comodules X and Y the tensor product of the cochain complexes is concentrated in degrees in 0, 1 and 2 and is given by

X^{1}⊗Y^{1} ⟵^{d^{1}} X^{1}⊗Y^{0} ⊕ X^{0}⊗Y^{1} ⟵^{d^{0}} X^{0}⊗Y^{0}

with d^{0} = \binom{d⊗1}{1⊗d} and d^{1} = (−1 ⊗ d, d ⊗ 1). Cotruncating this cochain complex we get the pair comodule

X ¯¯⊗Y = ((X ¯¯⊗Y)^{1} = ker(d^{1}) ⟵^{d} X^{0} ⊗ Y^{0} = (X ¯¯⊗Y)^{0})

with d induced by d_{0}. One readily checks the natural isomorphism

(2.2)  D(X ¯⊗Y) DX ¯¯⊗DY.

(2.3) Remark. Note that the full embedding of the category of pair comodules into the category of cochain complexes induced by the above identification has a right adjoint Tr^{∗} given by cotruncation: for a cochain complex

C^{∗} = (... ← C^{2} ⟵^{d^{1}} C^{1} ⟵^{d^{0}} C^{0} ⟵^{d^{−1}} C^{−1} ← ...) ,

one has

Tr^{∗}(C^{∗}) = (ker(d^{1}) ⟵^{d̄^{0}} C^{0}),

with d̄^{0} induced by d^{0}. Then clearly one has

X ¯¯⊗Y = Tr^{∗}(X ⊗ Y).

Using the fact that Tr^{∗} is a coreflection onto a full subcategory, one easily checks that the category Mod^{∗} together with the tensor product ¯ ¯⊗ and unit k^{∗} = (0 ← k) is a symmetric monoidal category, and Tr^{∗} is a monoidal functor.

We next consider the category Mod_{·}of graded modules, i. e. graded objects in Mod (graded k-vector spaces A_{·} = (A_{n})_{n∈} with lower indices which in each degree have finite dimension). For graded modules A_{·}, B_{·}we define their graded tensor product A_{·}⊗ B_{·}again in the usual way, i. e. by

(A_{·}⊗ B_{·})_{n} = ⨁_{i+j=n} A_{i} ⊗ B_{j}.

A graded pair comodule is a graded object of Mod^{∗}, i. e. a sequence X_{n} = (d_{n} : X_{n}^{0} → X_{n}^{1}) of pair comodules. We can also identify such a graded pair comodule X_{·} with the underlying morphism d of degree 0 between graded modules

X_{·}= (X_{·}^{1} ⟵^{d_{·}} X_{·}^{0}).

Now the tensor product X_{·}¯¯⊗Y_{·}of graded pair comodules X_{·}, Y_{·}is defined by

(2.4)  (X_{·}¯¯⊗Y_{·})_{n} = ⨁_{i+j=n} X_{i} ¯ ¯⊗Y_{j}.

This defines a monoidal structure on the category Mod_{·} of graded pair comodules. Morphisms in this category are of degree 0.

For two morphisms f, g : X_{·} → Y_{·} between graded pair comodules, a homotopy H : f ⇒ g is a morphism H : X_{·}^{1} → Y_{·}^{0} of degree 0 as in the diagram

(2.5)  [figure: X_{·}^{1}; H; f^{1}; g^{1}; Y_{·}^{1}; X_{·}^{0}; d; f^{0}; g^{0}; Y_{·}^{0},; d]

satisfying f^{0} − g^{0} = Hd and f^{1} − g^{1} = dH.

A pair coalgebra B_{·}is a comonoid in the monoidal category of graded pair comodules, with the diagonal

δ : B_{·}→ B_{·}¯¯⊗B_{·}.

We assume that B_{·}is concentrated in nonnegative degrees, that is B_{n} = 0 for n < 0.

Of course the duality functor D yields a duality functor

D : (Mod_{∗}^{·})^{op} → Mod_{·}^{∗}

which is compatible with the monoidal structure, i. e.

D(X^{·}¯⊗Y^{·}) (DX^{·}) ¯¯⊗(DY^{·}).

We also write D(X^{·}) = X_{·}.

More explicitly pair coalgebras can be described as follows.

(2.6) Definition. A pair coalgebra B_{·}is a graded pair comodule, i. e. an object

d_{·}: B_{·}^{0} → B_{·}^{1}

in Mod_{·}^{∗} with B_{n}^{1} = B_{n}^{0} = 0 for n < 0 such that B_{·}^{0} is a graded coalgebra in Mod_{·}, B_{·}^{1} is a graded B_{·}^{0}-B_{·}^{0}- bicomodule, and d_{·}is a bicomodule homomorphism. Moreover the diagram

\xymatrix{B_{·}^{1} \ar[r]^{λ} \ar[d]_{ρ} & B_{·}^{0} ⊗ B_{·}^{1} \ar[d]_{d_{·}⊗1} \\ B_{·}^{1} ⊗ B_{·}^{0} \ar[r]^{1⊗d_{·}} & B_{·}^{1} ⊗ B_{·}^{1}}

commutes, where λ, resp. ρ is the left, resp. right coaction.

It is easy to see that there results an exact sequence of graded B_{·}^{0}-B_{·}^{0}-bicomodules dual to (1.10)

(2.7)  0 ← π^{1}B_{·}← B_{·}^{1} ⟵^{d_{·}} B_{·}^{0} ← π^{0}B_{·}← 0

where in fact π^{0}B_{·}is a graded k-coalgebra, π^{1}B_{·}is a graded π^{0}B_{·}-π^{0}B_{·}-bicomodule, and B_{·}^{0} ← π^{0}B_{·}is a homomorphism of graded k-coalgebras.

One sees easily that the notions in this section correspond to those in the previous section under the duality functor D = Hom_{k}( , k). In particular, D carries (graded) pair algebras to (graded) pair coalgebras.

3. Folding systems

In this section we associate to a “right module system” M a category of M-algebras Alg_{M}^{r} which is a monoidal category if M is a “folding system”. Our main examples given by the -relation pair algebra ℛ of the Steenrod algebra 𝒜 and by the pair algebra ℬ of secondary cohomology operations are in fact comonoids in monoidal categories of such type, see sections 5 and 6. This generalizes the well known fact that the Steenrod algebra 𝒜 is a Hopf algebra, i. e. a comonoid in the category of algebras.

(3.1) Definition. Let A be a subcategory of the category of graded k-algebras. A right module system M over A is an assignment, to each A ∈ A, of a right A-module M(A), and, to each homomorphism f : A → A′ in A, of a homomorphism f_{∗} : M(A) → M(A′) which is f -equivariant, i. e.

f_{∗}(xa) = f_{∗}(x)f (a)

for any a ∈ A, x ∈ M(A). The assignment must be functorial, i. e. one must have (id_{A})_{∗} = id_{M(A)} for all A and (f g)_{∗} = f_{∗}g_{∗} for all composable f, g.

There are the obvious similar notions of a left module system and a bimodule system on a category of graded k-algebras A. Clearly any bimodule system can be considered as a left module system and a right module system by forgetting part of the structure.

(3.2) Examples. One obvious example is the bimodule system 𝟙 given by 𝟙(A) = A, f_{∗} = f for all A and f. Another example is the bimodule system Σ given by the suspension. That is, ΣA is given by the shift

Σ : A^{n−1} = (ΣA)^{n}

(n ∈ ) which is the identity map denoted by Σ. The bimodule structure for a, m ∈ A is given by

a(Σm) = (−1)^{deg(a)}Σ(am),
(Σm)a = Σ(ma).

We shall need the interchange of Σ which for graded modules U, V, W is the isomorphism

(3.3)  σ_{U,V,W}: U ⊗ (ΣV) ⊗ W ⟶^{}Σ(U ⊗ V ⊗ W)

which carries u ⊗ Σv ⊗ w to (−1)^{deg(u)}Σ(u ⊗ v ⊗ w).

Clearly a direct sum of module systems is again a module system of the same kind, so that in particular we get a bimodule system 𝟙 ⊕ Σ with (𝟙 ⊕ Σ)(A) = A ⊕ ΣA.

We are mainly interested in the bimodule system 𝟙 and the bimodule system 𝟙 ⊕ Σ which are in fact both folding systems, see (3.15) below.

(3.4) Definition. For a right module system M on the category of algebras A and an algebra A from A, an M-algebra of type A is a pair D_{∗} = (∂ : D_{1} → D_{0}) with π_{0}(D_{∗}) = A and π_{1}(D_{∗}) = M(A), such that D_{0} is a k-algebra, the quotient homomorphism D_{0} π_{0}D_{∗} = A is a homomorphism of algebras, D_{1} is a right D_{0}-module, ∂ is a homomorphism of right D_{0}-modules, and the induced structure of a right π_{0}(D_{∗})-module on π_{1}(D_{∗}) conicides with the original right A-module structure on M. For A, A′ in A, an M-algebra D_{∗} of type A, and another one D′_{∗} of type A′, a morphism D_{∗} → D′_{∗} of M-pair algebras is defined to be a commutative diagram of the form

\xymatrix{0 \ar[r] & M(A) \ar[r] \ar[d]_{f_{∗}} & D_{1} \ar[r]^{∂} \ar[d]^{f_{1}} & D_{0} \ar[r] \ar[d]^{f_{0}} & A \ar[r] \ar[d]^{f} & 0 \\ 0 \ar[r] & M(A′) \ar[r] & D′_{1} \ar[r]_{∂′} & D′_{0} \ar[r] & A′ \ar[r] & 0}

where f_{0} is a homomorphism of algebras and f_{1} is a right f_{0}-equivariant k-linear map. It is clear how to compose such morphisms, so that M-algebras form a category which we denote Alg_{M}^{r}.

With obvious modifications, we also get notions of M-algebra of type A when M is a left module system or a bimodule system; the corresonding categories of algebras will be denoted by Alg_{M}^{ℓ} and Alg_{M}^{b}, respectively. Moreover, for a bimodule system M there is also a further full subcategory

Alg_{M}^{pair} ⊂ Alg_{M}^{b}

whose objects, called M-pair algebras are those M-algebras which satisfy the pair algebra equation (∂x)y = x∂y for all x, y ∈ D_{1}.

(3.5) Remark. Note that if A contains k, then Alg_{M}^{?} has an initial object given by the M-algebra I = (0 : M(k) → k) of type k. Moreover if A contains the trivial algebra 0, then Alg_{M}^{?} also has a terminal object — the M-algebra 0 = M(0) → 0 of type 0. Here ? stands for ℓ, r or b if M is a left- right- or bimodule system, respectively.

(3.6) Definition. Let A be a category of graded algebras as above which in addition is closed under tensor product, i. e. k belongs to A and for any A, A′ from A the algebra A ⊗_{k} A′ also belongs to A. A right folding system on A is then defined to be a right module system M on A together with the collection of right A ⊗_{k} A′-module homomorphisms

λ_{A,A′}: A ⊗_{k} M(A′) → M(A ⊗_{k} A′),
ρ_{A,A′}: M(A) ⊗_{k} A′ → M(A ⊗_{k} A′)

for all A, A′ in A which are natural in the sense that for any homomorphisms f : A → A_{1}, f′: A′ → A′_{1} in A the diagrams

(3.7)  \xymatrix{A ⊗_{k} M(A′) \ar[r]^{λ_{A,A′}} \ar[d]_{f ⊗f′_{∗}} & M(A ⊗_{k} A′) \ar[d]^{(f ⊗f′)_{∗}} \\ A_{1} ⊗_{k} M(A′_{1}) \ar[r]^{λ_{A_{1},A′_{1}}} & M(A_{1} ⊗_{k} A′_{1})},  \xymatrix{M(A) ⊗_{k} A′ \ar[r]^{ρ_{A,A′}} \ar[d]_{f_{∗}⊗f′} & M(A ⊗_{k} A′) \ar[d]^{(f ⊗f′)_{∗}} \\ M(A_{1}) ⊗_{k} A′_{1} \ar[r]^{ρ_{A_{1},A′_{1}}} & M(A_{1} ⊗_{k} A′_{1})}

commute. Moreover the homomorphisms

(3.8)  \begin{aligned}λ_{k,A} & : k ⊗_{k} M(A) → M(k ⊗_{k} A), \\ ρ_{A,k} & : M(A) ⊗_{k} k → M(A ⊗_{k} k) \end{aligned}

must coincide with the obvious isomorphisms and the diagrams

\xymatrix{ & A ⊗_{k} M(A′⊗_{k} A′′) \ar[dr]^{λ_{A,A′⊗_{k}A′′}} \\ A ⊗_{k} A′⊗_{k} M(A′′) \ar[rr]^{λ_{A⊗_{k}A′,A′′}} \ar[ur]^{1⊗λ_{A′,A′′}} & & M(A ⊗_{k} A′⊗_{k} A′′),} (3.9)

\xymatrix{ & M(A ⊗_{k} A′) ⊗_{k} A′′ \ar[dr]^{ρ_{A⊗_{k}A′,A′′}} \\ M(A) ⊗_{k} A′⊗_{k} A′′ \ar[rr]^{ρ_{A,A′⊗_{k}A′′}} \ar[ur]^{ρ_{A,A′}A⊗1} & & M(A ⊗_{k} A′⊗_{k} A′′),} (3.10)

\xymatrix{ & M(A ⊗_{k} A′) ⊗_{k} A′′ \ar[dr]^{ρ_{A⊗_{k}A′,A′′}} \\ A ⊗_{k} M(A′) ⊗_{k} A′′ \ar[ur]^{λ_{A,A′}⊗1} \ar[dr]_{1⊗ρ_{A′,A′′}} & & M(A ⊗_{k} A′⊗_{k} A′′) \\ & A ⊗_{k} M(A′⊗_{k} A′′) \ar[ur]_{λ_{A,A′⊗_{k}A′′}}} (3.11)

must commute for all A, A′, A′′in A. A folding system is called symmetric if in addition the diagrams

\xymatrix{A ⊗_{k} M(A′) \ar[r]^{λ_{A,A′}} \ar[d]_{T_{A,M(A′)}} & M(A ⊗_{k} A′) \ar[d]^{M(T_{A,A′})} \\ M(A′) ⊗_{k} A \ar[r]^{ρ_{A′,A}} & M(A′⊗_{k} A)}

commute for all A, A′, where T is the graded interchange operator given in (1.5).

Once again, we have the corresponding obvious notions of a left folding system and a bifolding system.

For a right folding system M, the category Alg_{M}^{r} has a monoidal structure given by the folding product ˆ⊗ below. Given an M-algebra D of type A and another one, D′ of type A′, we define an M-pair algebra D ˆ⊗D′of type A ⊗ A′as the lower row in the diagram

(3.12)  [figure: 0; A⊗M(A′) ⊕ M(A)⊗A′; (λ_{A,A′},ρ_{A,A′}); push; (D ¯⊗D′)_{1}; ∂ ¯⊗; (D ¯⊗D′)_{0}; A ⊗ A′; 0; 0; M(A ⊗ A′); (D ˆ⊗D′)_{1}; ∂ ˆ⊗; D_{0} ⊗ D′_{0}; A ⊗ A′; 0.]

Here the leftmost square is required to be pushout, and the upper row is exact by (1.4).

(3.13) Proposition. For any right (resp. left, bi-) folding system M, the folding product defines a monoidal structure on Alg_{M}^{r} (resp. Alg_{M}^{ℓ}, Alg_{M}^{b}, Alg_{M}^{pair}), with unit object I = (0 : M(k) → k). If moreover the folding system is symmetric, then this monoidal structure is symmetric.

We only will use the monoidal categories Alg_{𝟙⊕Σ}^{r} and Alg_{𝟙}^{pair}.

Proof. To begin with, let us show that ˆ⊗ is functorial, i. e. let us for any morphisms f : D → E, f′: D′ → E′in Alg_{M}, define a morphism f ˆ⊗f′: D ˆ⊗E → D′ˆ⊗E′in a way compatible with identities and composition. We put (f ˆ⊗f′)_{0} = f_{0} ˆ⊗f′_{0} and define (f ˆ⊗f′)_{1} as the unique homomorphism making the following diagram commute:

\xymatrix{B⊗M(B′) ⊕ M(B)⊗B′ \ar[rrr] \ar[ddd]_{(λ_{B,B′},ρ_{B,B′})} &  &  & (E ¯⊗E′)_{1} \ar[ddd] \\  & A⊗M(A′) ⊕ M(A)⊗A′ \ar[r] \ar[d]_{(λ_{A,A′},ρ_{A,A′})} \ar[ul]_{f⊗f′_{∗}⊕f_{∗}⊗f′} & (D ¯⊗D′)_{1} \ar[d] \ar[ur]^{(f ¯⊗f′)_{1}} \\  & M(A ⊗ A′) \ar[r] \ar[dl]^{(f ⊗f′)_{∗}} & (D ˆ⊗D′)_{1} \ar@{-->}[dr]_{(f ˆ⊗f′)_{1}} \\ M(B ⊗ B′) \ar[rrr] &  &  & (E ˆ⊗E′)_{1}}

where the left hand trapezoid commutes by (3.7). Using universality of pushout it is clear that right equivariance of f_{1} and f′_{1} implies that of (f ˆ⊗f′)_{1} so that this indeed defines a morphism in Alg_{M}. The same universality implies compatibility with composition.

Next to show that I = (0 : M(k) → k) is a unit object first note that for an M-algebra D by (1.3) one has

I ¯⊗D = Tr_{∗}(M(k) ⊗ D_{1} \underrightarrow{\binom{0}{1⊗∂}} D_{1} ⊕ M(k)⊗D_{0} ⟶^{(∂,0)} D_{0}) (D_{1} ⊕ M(k)⊗A ⟶^{(∂,0)} D_{0}).

From this using (3.8) it is easy to see that (I ˆ⊗D)_{1} is given by the pushout

\xymatrix{M(A) ⊕ M(k)⊗A \ar[r]^{incl⊕1} \ar[d]_{proj} & D_{1} ⊕ M(k)⊗A \ar[d] \\ M(A) \ar[r] & (I ˆ⊗D)_{1}}

so that there is a canonical isomorphism (I ˆ⊗D)_{1} D_{1} compatible with the canonical isomorphism k⊗D_{0} D_{0}. Symmetrically one constructs the isomorphism D ˆ⊗I D.

Turning now to associativity, first note that the tensor product (1.2)can be equivalently stated as defining (D ¯⊗D′)_{1} by the requirement that the diagram

[figure: D_{1} ⊗ D′_{1}; push; D_{0} ⊗ D′_{1}; D_{1} ⊗ D′_{0}; (D ¯⊗D′)_{1}]

be pushout. Then combining diagrams we see that (D ˆ⊗D′)_{1} can be equivalently defined as the colimit of the following diagram:

(3.14)  [figure: D_{0} ⊗ M(A′); D_{1} ⊗ D′_{1}; M(A) ⊗ D′_{0}; D_{0} ⊗ D′_{1}; M(A ⊗ A′); D_{1} ⊗ D′_{0}]

where the map D_{0} ⊗ M(A′) → M(A ⊗ A′) is the composite D_{0} ⊗ M(A′) → A ⊗ M(A′) → M(A ⊗ A′) and similarly for M(A) ⊗ D′_{0} → M(A ⊗ A′). Hence ((D ˆ⊗D′) ˆ⊗D′′)_{1} is given by the colimit of the diagram

[figure: D_{0} ⊗ D′_{0} ⊗ M(A′′); (D ˆ⊗D′)_{1} ⊗ D′′_{1}; M(A ⊗ A′) ⊗ D′′_{0}; D_{0} ⊗ D′_{0} ⊗ D′′_{1}; M(A ⊗ A′⊗ A′′); (D ˆ⊗D′)_{1} ⊗ D′′_{0}.]

Substituting here the diagram for (D ˆ⊗D′)_{1} we obtain that this is the same as colimit of a diagram of the form

\xymatrix{ &  &  &  & D_{0} ⊗ D′_{1} ⊗ D′′_{0} \\  &  & D_{0} ⊗ D′_{1} ⊗ D′′_{1} \ar[dll] \ar[urr] & D_{0} ⊗ M(A′) ⊗ D′′_{0} \ar[ur] \ar[dl] \\ D_{0} ⊗ D′_{0} ⊗ D′′_{1} & D_{0} ⊗ D′_{1} ⊗ M(A′′) \ar[l] \ar[r] & M(A ⊗ A′⊗ A′′) &  & D_{1} ⊗ D′_{1} ⊗ D′′_{0} \ar[uu] \ar[dd] \\  &  & D_{1} ⊗ D′_{0} ⊗ D′′_{1} \ar[ull] \ar[drr] & M(A) ⊗ D′_{0} ⊗ D′′_{0} \ar[ul] \ar[dr] \\  &  &  &  & D_{1} ⊗ D′_{0} ⊗ D′′_{0}.}

Treating now (D ˆ⊗(D′ˆ⊗D′′))_{1} in the same way we obtain that it is colimit of a diagram with same objects; then, using (3.9), (3.11), and (3.10), one can see that also morphisms in these diagrams are the same.

Finally suppose that M is a symmetric folding system. Then for any M-algebras D, D′ of type A, A′ respectively, there is a commutative diagram

\xymatrix{ &  &  & M(A ⊗ A′) \ar[d] \\  &  &  & M(A′⊗ A) \\  & D_{0} ⊗ M(A′) \ar[uurr] \ar[dddl] \ar[dr] &  &  &  & M(A) ⊗ D′_{0} \ar[uull] \ar[dddr] \ar@{-->}[dl] \\  &  & M(A′) ⊗ D_{0} \ar[uur] \ar[dl] &  & D′_{0} ⊗ M(A) \ar[uul] \ar[dr] \\  & D′_{1} ⊗ D_{0} &  & D′_{1} ⊗ D_{1} \ar[ll] \ar[rr] &  & D′_{0} ⊗ D_{1} \\ D_{0} ⊗ D′_{1} \ar@{-->}[ur] &  &  & D_{1} ⊗ D′_{1} \ar[u] \ar[lll] \ar[rrr] &  &  & D_{1} ⊗ D′_{0} \ar@{-->}[ul]}

which induces a map from the colimit of the outer triangle to that of the inner one, i. e. by (3.14) a map (D ˆ⊗D′)_{1} → (D′ˆ⊗D)_{1}. It is then straightforward to check that this defines an interchange for the monoidal structure.

(3.15) Examples. The bimodule system 𝟙 above clearly has a structure of a folding system, with λ and ρ both identity maps. Also the bimodule system 𝟙 ⊕ Σ is a folding system via the obvious isomorphisms

(3.16)  λ_{A,A′}: A ⊗ (A′⊕ ΣA′) A⊗A′⊕ A⊗ΣA′ ⟶^{1⊕σ} A⊗A′⊕ Σ(A⊗A′),
(3.17)  ρ_{A,A′}: (A ⊕ ΣA) ⊗ A′ A⊗A′⊕ (ΣA)⊗A′ A⊗A′⊕ Σ(A⊗A′)

where in (3.16), the interchange (3.3) for Σ is used.

(3.18) Lemma. The isomorphisms (3.16), (3.17) equip the bimodule system 𝟙 ⊕ Σ with the structure of a symmetric folding system on any category A of algebras closed under tensor products.

Proof. It is obvious that 𝟙 with the identity maps is a folding system, and that a direct sum of folding systems is a folding system again, so it suffices to show that Σ is a folding system.

The right diagram in (3.7) is trivially commutative, while commutativity of the left one follows from

σ_{A_{1},M(A′_{1})}(f (a) ⊗ Σf′(a′)) = (−1)^{deg(a)}Σ(f (a) ⊗ f′(a′))

= Σ(f ⊗ f′)((−1)^{deg(a)}Σ(a ⊗ a′)) = Σ(f ⊗ f′)σ_{A,M(A′)}(a ⊗ Σa′)

for any a ∈ A, a′ ∈ A′, f : A → A_{1}, f′: A′ → A′_{1}. Next, the diagrams (3.8)commute since k is concentrated in degree 0.

The diagrams (3.10) commute trivially, as only right actions are involved. Commutativity of (3.9) follows from the obvious equality

(−1)^{deg(a)}Σ(a ⊗ (−1)^{deg(a′)}a′⊗ a′′) = (−1)^{deg(a⊗a′)}Σ(a ⊗ a′⊗ a′′)

and that of (3.11) is also obvious from

\xymatrix{ & (−1)^{deg(a)}Σ(a ⊗ a′) ⊗ a′′ \ar@{|->}[dr] \\ a ⊗ Σ(a′) ⊗ a′′ \ar@{|->}[ur] \ar@{-->}[dr] &  & (−1)^{deg(a)}Σ(a ⊗ a′⊗ a′′) \\  & a ⊗ Σ(a′⊗ a′′) \ar@{|->}[ur]}

Thus by (3.13) the folding system 𝟙 ⊕ Σ yields a well-defined monoidal category Alg_{𝟙⊕Σ}^{r} of 𝟙 ⊕ Σalgebras as in (3.4). The initial object and at the same time the unit for the monoidal structure of Alg_{𝟙⊕Σ}^{r} is by (3.5) and (3.13)

I_{𝟙⊕Σ} = ( ⊕ Σ ⟶^{0} ) .

For Alg_{𝟙}^{r} it is

I_{𝟙} = ( ⟶^{0} ) .

The projections q : A ⊕ ΣA → A can be used to construct a monoidal functor

(3.19)  q : Alg_{𝟙⊕Σ}^{r} → Alg_{𝟙}^{r}

carrying an object D in Alg_{𝟙⊕Σ}^{r} to the pushout in the following diagram

[figure: A ⊕ ΣA; q; push; D_{1}; D_{0}; A; A; q(D)_{1}; q(D)_{0}; A.]

Evidently q(I_{𝟙⊕Σ} ) = I_{𝟙}.

4. Unfolding systems

It is clear how to dualize the constructions from the previous section along the lines of section 2. We will not give detailed definitions but only briefly indicate the underlying structures.

We thus consider a category C of graded k-coalgebras, and define a right comodule system N on C as an assignment, to each coalgebra C in C, of a C-comodule N(C), and to each homomorphism f : C → C′ of coalgebras of an f -equivariant homomorphism f_{∗} : N(C) → N(C′), i. e. the diagram

\xymatrix{N(C) \ar[r]^{coaction} \ar[d]_{f_{∗}} & N(C) ⊗ C \ar[d]^{f_{∗}⊗f} \\ N(C′) \ar[r]^{coaction} & N(C′) ⊗ C′}

is required to commute. Similarly one defines left comodule systems and bicomodule systems. As before, we have a bicomodule system 𝟙 given by 𝟙(C) = C and also Σ, 𝟙 ⊕ Σ defined dually to (3.2).

Then further for a right comodule system N on C and for a coalgebra C from C one defines an Ncoalgebra of type C by dualizing (3.4). It is thus a pair D^{∗} = (d : D^{0} → D^{1}) where D^{0} is a coalgebra, D^{1} is a right D^{0}-comodule and d is a comodule homomorphism. Moreover one must have π^{0}(D^{∗}) = C, π^{1}(D^{∗}) = N(C), and the C-comodule structure on N(C) induced by this must be the one coming from the comodule system N. With morphisms defined dually to (3.4), the N-coalgebras form a category Coalg_{N}^{r}. Similarly one defines categories Coalg_{N}^{ℓ} and Coalg_{N}^{pair} ⊂ Coalg_{N}^{b} for a left, resp. bicomodule system N. These categories have the initial object 0 : 0 → N(0) and the terminal object 0 : k → N(k).

Also dually to (3.6) one defines unfolding systems as comodule systems N equipped with C ⊗ C′- comodule homomorphisms

l^{C,C′}: N(C ⊗ C′) → C ⊗ N(C′)
r^{C,C′}: N(C ⊗ C′) → N(C) ⊗ C′

for all C, C′ ∈ C required to satisfy obvious duals to the diagrams (3.7) – (3.11). Also there is an obvious notion of a symmetric unfolding system.

Then for an unfolding system N we can dualize (3.12) to obtain definition of the unfolding product D ˇ⊗D′of N-coalgebras via the upper row in the diagram

[figure: 0; C ⊗ C′; D^{0} ⊗ D′^{0}; d ˇ⊗; (D ˇ⊗D′)^{1}; pull; N(C ⊗ C′); _{\binom{l^{C,C′}}{r^{C,C′}}}; 0; 0; C ⊗ C′; (D ¯¯⊗D′)^{0}; d^{¯¯⊗}; (D ¯¯⊗D′)^{1}; C⊗N(C′) ⊕ N(C)⊗C′; 0]

where now the rightmost square is required to be pullback and the lower row is exact by the dual of (1.4).

It is then straightforward to dualize (3.13), so we conclude that for any unfolding system N the unfolding product equips the category Coalg_{N}^{?} with the structure of a monoidal category, symmetric if N is symmetric. Here, “?” stands for “r”, “l”, “b” or “pair”, according to the type of N. Obviously also the dual of (3.18) holds, so that the categories Coalg_{𝟙}^{pair} and Coalg_{𝟙⊕Σ}^{r} have monoidal structures given by the unfolding product.

5. The -relation pair algebra of the Steenrod algebra

Fix a prime p and let = /p^{2} be the ring of integers mod p^{2}, with the quotient map = _{p} = /p . Let 𝒜 be the mod p Steenrod algebra and let

E_{𝒜} = \begin{cases}\{Sq^{1}, Sq^{2}, ...\} & for p = 2, \\ \{P^{1}, P^{2}, ...\}∪ \{β, βP^{1}, βP^{2}, ...\} & for odd p \end{cases}

be the set of generators of the algebra 𝒜. We consider the following algebras and homomorphisms

(5.1)  [figure: q : ℬ_{0}; ℱ_{0}; q_{ℱ}; 𝒜; T_{}(E_{𝒜} ); T_{}(E_{𝒜} ); .]

Here T_{k}(S) for a commutative ring k denotes the free associative k-algebra with unit generated by the set S, i. e. the tensor algebra of the free k-module on S. The map q_{ℱ} is the algebra homomorphism which is the identity on E_{𝒜} . For f ∈ ℱ_{0} we denote the element q_{ℱ} (f) ∈ 𝒜 by

¯f̄ = q_{ℱ} (f).

Let R_{ℬ} denote the kernel of q, i. e. there is a short exact sequence

[figure: R_{ℬ}; ℬ_{0}; q; 𝒜 .]

This short exact sequence gives a long exact sequence

[figure: Tor(R_{ℬ} , ); Tor(ℬ_{0}, ); Tor(𝒜 , ); i; R_{ℬ} ⊗; ℬ_{0} ⊗; 𝒜 ⊗ .]

Here A ⊗ A/pA and Tor(A, ) is just the p-torsion part of A for an abelian group A, so the connecting homomorphism i sends a = q(b) + pℬ_{0} to pb + pR_{ℬ} . It follows that the second homomorphism in the above sequence is zero. Moreover clearly we can identify ℬ_{0} ⊗ = ℱ_{0} and Tor(𝒜 , ) = 𝒜, so that there is an exact sequence

(5.2)  [figure: 𝒜; i; ℛ_{1}^{}; ∂; ℛ_{0}^{}; 𝒜; R_{ℬ} ⊗; ℱ_{0}]

One has

(5.3) Lemma. The pair ℛ^{}= (∂ : ℛ_{1}^{}→ ℛ_{0}^{}) above has a pair algebra structure compatible with the standard bimodule structure of 𝒜 on itself, so that ℛ^{}yields an object in Alg_{𝟙}^{pair}, see (3.4).

Proof. Clearly mod p reduction of any pair algebra over is a pair algebra over . Then let ℛ^{}be the mod p reduction of the pair algebra R_{ℬ} ℬ_{0}. Thus the ℱ_{0}-ℱ_{0}-bimodule structure on ℛ_{1}^{}= R_{ℬ} /pR_{ℬ} is just the mod p reduction of the ℬ_{0}-ℬ_{0}-bimodule structure on R_{ℬ} , i. e. b′+ pℬ_{0} ∈ ℛ_{0}^{}= ℬ_{0}/pℬ_{0} acts on r + pR_{ℬ} ∈ ℛ_{1}^{}= R_{ℬ} /pR_{ℬ} via

(b′+ pℬ_{0})(r + pR_{ℬ} ) = b′r + pR_{ℬ} .

Moreover the above inclusion 𝒜 R_{ℬ} /pR_{ℬ} sends an element q(b) to pb + pR_{ℬ} . Then the action of a′ = q(b′) ∈ 𝒜 on i(a) = pb + pR_{ℬ} ∈ i(𝒜) = ker∂ induced by this pair algebra is given as follows:

a′i(a) = q_{ℱ} (b′+ pℬ_{0})(pb + pR_{ℬ} ) = pb′b + pR_{ℬ} = iq(b′b) = i(a′a)

and similarly for the right action.

We call the object ℛ^{}of the category Alg_{𝟙}^{pair} the -relation pair algebra of 𝒜.

(5.4) Theorem. The 𝟙-pair algebra ℛ^{}has a structure of a cocommutative comonoid in the symmetric monoidal category Alg_{𝟙}^{pair}.

Proof. For n 0, let R_{ℬ}^{(n)} denote the kernel of the map q^{⊗n}, so that there is a short exact sequence

[figure: R_{ℬ}^{(n)}; ℬ_{0}^{⊗n}; q^{⊗n}; 𝒜^{⊗n}]

and similarly to (5.3) there is a pair algebra of the form

[figure: 𝒜^{⊗n}; R_{ℬ}^{(n)} ⊗; ℱ_{0}^{⊗n}; q_{ℱ}^{⊗n}; 𝒜^{⊗n}]

determining an object ℛ^{(n)} in Alg_{𝟙}^{pair}. Then one has the following lemma which yields natural examples of folding products in Alg_{𝟙}^{pair}.

(5.5) Lemma. There is a canonical isomorphism ℛ^{(n)} (ℛ^{})^{ˆ⊗n} in Alg_{𝟙}^{pair}.

Proof. Using induction, we will assume given an isomorphism α_{n} : (ℛ^{})^{ˆ⊗n} ℛ^{(n)} and construct α_{n+1} in a canonical way. To do this it clearly suffices to construct a canonical isomorphism ℛ^{}ˆ⊗ℛ^{(n)} ℛ^{(n+1)} as then its composite with ℛ^{}ˆ⊗α_{n} will give α_{n+1}.

To construct a map (ℛ^{}ˆ⊗ℛ^{(n)})_{1} → ℛ_{1}^{(n+1)} means by (3.14) the same as to find three dashed arrows making the diagram

[figure: R_{ℬ} ⊗ R_{ℬ}^{(n)} ⊗; ℱ_{0} ⊗ R_{ℬ}^{(n)}; R_{ℬ} ⊗ ℱ_{0}^{⊗n}; R_{ℬ}^{(n+1)} ⊗; ℱ_{0} ⊗ 𝒜^{⊗n}; 𝒜^{⊗(n+1)}; 𝒜 ⊗ ℱ_{0}^{⊗n}]

commute. For this we use the commutative diagram

[figure: R_{ℬ} ⊗ R_{ℬ}^{(n)}; ℬ_{0} ⊗ R_{ℬ}^{(n)}; R_{ℬ} ⊗ ℬ_{0}^{⊗n}; R_{ℬ}^{(n+1)}; ℬ_{0} ⊗ 𝒜^{⊗n}; 𝒜^{⊗(n+1)}; 𝒜 ⊗ ℬ_{0}^{⊗n};]

This diagram has a commutative subdiagram

[figure: p(R_{ℬ} ⊗ R_{ℬ}^{(n)}); pℬ_{0} ⊗ R_{ℬ}^{(n)}; R_{ℬ} ⊗ pℬ_{0}^{⊗n}; pR_{ℬ}^{(n+1)}; pℬ_{0} ⊗ 𝒜^{⊗n}; 0; 𝒜 ⊗ pℬ_{0}^{⊗n};]

It is obvious that quotient by this subdiagram gives us a diagram of the kind we need.

We thus obtain a map (ℛ^{}ˆ⊗ℛ^{(n)})_{1} → R_{ℬ}^{(n+1)} ⊗ . Moreover by its construction this map fits into the commutative diagram

[figure: 𝒜^{⊗(n+1)}; (ℛ^{}ˆ⊗ℛ^{(n)})_{1}; ℱ_{0}^{⊗(n+1)}; 𝒜^{⊗(n+1)}; 𝒜^{⊗(n+1)}; R_{ℬ}^{(n+1)} ⊗; ℱ_{0}^{⊗(n+1)}; 𝒜^{⊗(n+1)}]

with exact rows, hence by the five lemma it is an isomorphism.

Using the lemma we next construct the diagonal of ℛ^{}given by

[figure: R_{ℬ} ⊗; Δ^{}⊗1; ℛ_{1}^{}; ∂; Δ; ℛ_{0}^{}; Δ; ℱ_{0}; Δ; R_{ℬ}^{(2)} ⊗; ; (ℛ^{}ˆ⊗ℛ^{})_{1}; ∂ ˆ⊗; ℛ_{0}^{}⊗ ℛ_{0}^{}; ℱ_{0} ⊗ ℱ_{0}.]

Here Δ^{}is defined by the commutative diagram

(5.6)  [figure: R_{ℬ}; Δ^{}; ℬ_{0}; Δ^{}; R_{ℬ}^{(2)}; ℬ_{0} ⊗ ℬ_{0},]

where the diagonal Δ^{}on ℬ_{0} is defined on generators by

Δ^{}(Sq^{n}) = ∑_{i=0}^{n} Sq^{i} ⊗ Sq^{n−i}  for p = 2,

\begin{matrix}Δ^{}(β) & = β ⊗ 1 + 1 ⊗ β, \\ Δ^{}(P^{n}) & = ∑_{i+j=n} P^{i} ⊗ P^{j}, \\ Δ^{}(P_{β}^{n}) & = ∑_{i+j=n} (P_{β}^{i} ⊗ P^{j} + P^{i} ⊗ P_{β}^{j}) \end{matrix} \}for odd p

(with Sq^{0} = 1, P^{0} = 1 as usual) and extended to the whole ℬ_{0} as the unique algebra homomorphism with respect to the algebra structure on ℬ_{0} ⊗ ℬ_{0} given by the nonstandard interchange formula

[figure: ℬ_{0} ⊗ ℬ_{0} ⊗ ℬ_{0} ⊗ ℬ_{0}; μ⊗μ; ℬ_{0} ⊗ ℬ_{0} ⊗ ℬ_{0} ⊗ ℬ_{0}; 1⊗T^{}⊗1; μ_{⊗}; ℬ_{0} ⊗ ℬ_{0}]

with

T^{}: ℬ_{0} ⊗ ℬ_{0} ⟶^{}ℬ_{0} ⊗ ℬ_{0}

T^{}(x ⊗ y) = (−1)^{p deg(x) deg(y)}y ⊗ x.

In particular, clearly for all p one has T^{}Δ^{}= Δ^{}, i. e. the coalgebra structure on ℬ_{0} is cocommutative. The counit for ℛ^{}is given by the diagram

(5.7)  [figure: 𝒜; ϵ; R_{ℬ} ⊗; ℱ_{0}; ϵ; 𝒜; ϵ; ; ; 0; ; ]

where the map R_{ℬ} ⊗ → sends the generator (p1_{ℬ_{0}} ) ⊗ 1 in degree 0 to 1 and all elements in higher degrees to zero. It is then clear from the formula for Δ^{}that this indeed gives a counit for this diagonal.

Finally to prove coassociativity, by the lemma it suffices to consider the diagram

\xymatrix{ &  & R_{ℬ} \ar@{^{(}->}[d] \ar[ddll]_{Δ^{}} \ar[ddrr]^{Δ^{}} \\  &  & ℬ_{0} \ar[dl]_{Δ^{}} \ar[dr]^{Δ^{}} \\ R_{ℬ}^{(2)} \ar@{^{(}->}[r] \ar[ddrr] & ℬ_{0}^{⊗2} \ar[dr]_{1⊗Δ^{}} &  & ℬ_{0}^{⊗2} \ar[dl]^{Δ^{}⊗1} & R_{ℬ}^{(2)} \ar[ddll] \ar@{^{(}->}[l] \\  &  & ℬ_{0}^{⊗3} \\  &  & R_{ℬ}^{(3)} \ar@{^{(}->}[u]}

6. The algebra of secondary cohomology operations

Let us next consider a derivation of degree 0 of the form

: 𝒜 → Σ𝒜 ,

uniquely determined by

(6.1)

Sq^{n} = ΣSq^{n−1} for p = 2,
\begin{matrix}β & = Σ1, \\ (P^{i}) & = 0, i 0 \end{matrix} \}for odd p.

We will use to define an 𝒜 -𝒜 -bimodule

𝒜 ⊕_{}Σ𝒜

as follows. The right 𝒜 -module structure is the same as on 𝒜 ⊕ Σ𝒜 above, i. e. one has (x, Σy)a = (xa, Σya). As for the left 𝒜 -module structure, it is given by

a(x, Σy) = (ax, (−1)^{deg(a)}Σay + (a)x).

There is a short exact sequence of 𝒜 -𝒜 -bimodules

0 → Σ𝒜 → 𝒜 ⊕_{}Σ𝒜 → 𝒜 → 0

given by the standard inclusion and projection.

(6.2) Remark. The above short exact sequence of bimodules and the derivation correspond to each other under the well known description of the first Hochschild cohomology group by bimodule extensions and derivations, respectively. Indeed, more generally recall that for a graded k-algebra A and an A-A-bimodule M, one of the possible definitions of the Hochschild cohomology of A with coefficients in M is

HH^{n}(A; M) = Ext_{A⊗_{k}A^{∘}}^{n}(A, M).

On the other hand, HH^{1}(A; M) can be also described in terms of derivations. Recall that an M-valued derivation on A is a k-linear map : A → M of degree 0 satisfying

(xy) = (x)y + (−1)^{deg(x)}x (y)

for any x, y ∈ A. Such derivations form a k-vector space Der(A; M). A derivation = ι_{m} is called inner if there is an m ∈ M such that

(x) = mx − (−1)^{deg(x)}xm = ι_{m}(x)

for all x ∈ A. These form a subspace Ider(A; M) ⊂ Der(A; M) and one has an isomorphism HH^{1}(A; M) Der(A; M)/ Ider(A; M). Moreover there is an exact sequence

0 → HH^{0}(A; M) → M ⟶^{ι} Der(A; M) → HH^{1}(A; M) → 0.

Explicitly, the isomorphism

Der(A; M)/ Ider(A; M) Ext_{A⊗A^{∘}}^{1}(A, M),

can be described by assigning to a class of a derivation : A → M the class of the extension

0 → M → A ⊕_{}M → A → 0

where as a vector space, A ⊕_{}M = A ⊕ M, the maps are the canonical inclusion and projection and the bimodule structure is given by

a(x, m)= (ax, am + (a)x),
(x, m)a= (xa, ma).

Obviously the 𝒜 ⊕_{}Σ𝒜 above is an example of this construction.

(6.3) Definition. A Hopf pair algebra 𝒱 (associated to 𝒜) is a pair algebra ∂ : 𝒱_{1} → 𝒱_{0} over together with the following commutative diagram in the category of ℱ_{0}-ℱ_{0}-bimodules

(6.4)  [figure: Σ𝒜; Σ𝒜; 𝒜 ⊕_{}Σ𝒜; q; 𝒱_{1}; ∂; q; 𝒱_{0}; 𝒜; 𝒜; ℛ_{1}^{}; ℛ_{0}^{}; 𝒜]

with exact rows and columns. The pair morphism q : 𝒱 → ℛ^{}will be called the -structure of 𝒱 . Moreover 𝒱 has a structure of a comonoid in Alg_{𝟙⊕Σ}^{r} and q is compatible with the Alg_{𝟙}^{pair}-comonoid structure on ℛ^{}in (5.4), in the sense that the diagrams

(6.5)  \xymatrix{𝒱_{1} \ar[r]^{Δ_{𝒱}} \ar[d]_{q} & (𝒱 ˆ⊗𝒱 )_{1} \ar[d]^{q ˆ⊗q} \\ ℛ_{1}^{} \ar[r]^{Δ_{ℛ}} & (ℛ^{}ˆ⊗ℛ^{})_{1}}

and

(6.6)  \xymatrix{𝒱_{1} \ar[d]_{q} \ar[r]^{ϵ_{𝒱}} & ⊕ Σ \ar[d] \\ ℛ_{1}^{} \ar[r]^{ϵ_{ℛ}} & }

commute.

We next observe that the following diagrams commute:

[figure: 𝒜; ; δ; Σ𝒜; Σδ; 𝒜 ⊗ 𝒜; ⊗1; Σ𝒜 ⊗ 𝒜; Σ(𝒜 ⊗𝒜),; 𝒜; ; δ; Σ𝒜; Σδ; 𝒜 ⊗ 𝒜; 1⊗; 𝒜 ⊗ Σ𝒜; σ; Σ(𝒜 ⊗𝒜)]

where σ is the interchange for Σ in (3.3). Or, on elements,

(6.7)  ∑ (a_{ℓ}) ⊗ a_{r} = ∑ (a)_{ℓ} ⊗ (a)_{r} = ∑ σ(a_{ℓ} ⊗ (a_{r})),

where we use the Sweedler notation for the diagonal

δ(x) = ∑ x_{ℓ} ⊗ x_{r}.

(6.8) Remark. The above identities have a simple explanation using dualization. We will see in (9.7) below that the map dual to is the map Σ𝒜_{∗} → 𝒜_{∗} given, for p = 2, by multiplication with the degree 1 generator ζ_{1} ∈ 𝒜_{∗} and for odd p by the degree 1 generator τ_{0}. Then the duals of (6.7) are the obvious identities for any x, y ∈ 𝒜_{∗}

(ζ_{1}x)y = ζ_{1}(xy) = x(ζ_{1}y)

for p = 2 and

(τ_{0}x)y = τ_{0}(xy) = (−1)^{deg(x)}x(τ_{0}y)

for odd p (recall that 𝒜_{∗} is graded commutative).

Using (6.7) we prove:

(6.9) Lemma. For a Hopf pair algebra 𝒱 there is a unique left action of ℱ_{0} on (𝒱 ˆ⊗𝒱 )_{1} such that the quotient map

(𝒱 ¯⊗𝒱 )_{1} (𝒱 ˆ⊗𝒱 )_{1}

is ℱ_{0}-equivariant. Here we use the pair algebra structure on 𝒱 ¯⊗𝒱 to equip (𝒱 ¯⊗𝒱 )_{1} with an ℱ_{0} ⊗ ℱ_{0}- bimodule structure and then turn it into a left ℱ_{0}-module via restriction of scalars along Δ : ℱ_{0} → ℱ_{0} ⊗ ℱ_{0}.

Proof. Uniqueness is clear as the module structure on the quotient of any module M by a submodule is clearly uniquely determined by the module structure on M.

For the existence, consider the diagram

(6.10)  [figure: ℱ_{0} ⊗ (𝒜 ⊕_{}Σ𝒜); 𝒱_{1} ⊗ 𝒱_{1}; (𝒜 ⊕_{}Σ𝒜) ⊗ ℱ_{0}; ℱ_{0} ⊗ 𝒱_{1}; 𝒜 ⊗𝒜 ⊕_{⊗1} Σ(𝒜 ⊗𝒜); 𝒱_{1} ⊗ ℱ_{0}.]

whose colimit, by (3.14), is (𝒱 ˆ⊗𝒱 )_{1}, with the right ℱ_{0} ⊗ ℱ_{0}-module structure coming from the category Alg_{𝟙⊕Σ}^{r} . It then suffices to show that all maps in this diagram are also left ℱ_{0}-equivariant, if one uses the left ℱ_{0}-module structure by restricting scalars along the diagonal ℱ_{0} → ℱ_{0} ⊗ ℱ_{0}.

This is trivial except possibly for two of the maps involved. For the map

Φ : ℱ_{0} ⊗ (𝒜 ⊕_{}Σ𝒜) → 𝒜 ⊗𝒜 ⊕_{⊗𝒜} Σ(𝒜 ⊗𝒜)

given by

Φ(f′⊗ (x, Σy)) = ( ¯f̄′ ⊗ x, (−1)^{deg(f′)}Σ ¯f̄′ ⊗ y),

this amounts to checking that for any f, f′ ∈ ℱ_{0} and x, y ∈ 𝒜 one must have

∑ (f_{ℓ} ⊗ f_{r})( ¯f̄′ ⊗ x, (−1)^{deg(f′)}Σ ¯f̄′ ⊗ y) = Φ((−1)^{deg(f_{r}) deg(f′)} ∑ f_{ℓ}f′⊗ ( ¯f̄_{r}x, (−1)^{deg(f_{r})}Σ ¯f̄_{r}y + ( ¯f̄_{r})x)),

where again the above Sweedler notation

Δ(f) = ∑ f_{ℓ} ⊗ f_{r},

is used for the diagonal of ℱ_{0} too, and ¯ f̄′ denotes q_{ℱ} (f′) by the notation in (5.1).

The left hand side expression then expands as

∑ ((−1)^{deg(f_{r}) deg(f′)} ¯ f̄_{ℓ} ¯f̄′ ⊗ ¯ f̄_{r}x,

(−1)^{deg(f_{r}) deg(f′)} (−1)^{deg(f)}(−1)^{deg(f′)}Σ ¯f̄_{ℓ} ¯f̄′ ⊗ ¯ f̄_{r}y + (−1)^{deg(f_{r}) deg(f′)} ( ¯f̄_{ℓ}) ¯f̄′ ⊗ ¯ f̄_{r}x)

and the right hand side expands as

(−1)^{deg(f_{r}) deg(f′)} ∑ ( ¯f̄_{ℓ} ¯f̄′ ⊗ ¯ f̄_{r}x, (−1)^{deg(f_{ℓ}f′)} ((−1)^{deg(f_{r})}Σ ¯f̄_{ℓ} ¯f̄′ ⊗ ¯ f̄_{r}y + ¯ f̄_{ℓ} ¯f̄′ ⊗ ( ¯f̄_{r})x)).

Thus left equivariance of Φ is equivalent to the equality

∑ ( ¯f̄_{ℓ}) ¯f̄′ ⊗ ¯ f̄_{r}x = ∑ (−1)^{deg(f_{ℓ}f′)} ¯ f̄_{ℓ} ¯f̄′ ⊗ ( ¯f̄_{r})x.

This is easily deduced from

∑ ( ¯f̄_{ℓ}) ⊗ ¯ f̄_{r} = ∑ (−1)^{deg(f_{ℓ})} ¯f̄_{ℓ} ⊗ ( ¯f̄_{r}),

which is an instance of (6.7).

For another map

Ψ : (𝒜 ⊕_{}Σ𝒜) ⊗ ℱ_{0} → 𝒜 ⊗𝒜 ⊕_{⊗𝒜} Σ(𝒜 ⊗𝒜)

given by

Ψ((x, Σy) ⊗ f′) = (x ⊗ ¯ f̄′, Σy ⊗ ¯ f̄′)

the equality to check is

∑ (f_{ℓ} ⊗ f_{r})(x ⊗ ¯ f̄′, Σy ⊗ ¯ f̄′) = Ψ((−1)^{deg(f_{r}) deg(x,Σy)} ∑ ( ¯f̄_{ℓ}x, (−1)^{deg(f_{ℓ})}Σ ¯f̄_{ℓ}y + ( ¯f̄_{ℓ})x) ⊗ f_{r}f′).

Here the left hand side expands as

∑ ((−1)^{deg(f_{r}) deg(x)} ¯f̄_{ℓ}x ⊗ ¯ f̄_{r} ¯f̄′, (−1)^{deg(f_{r}) deg(Σy)}(−1)^{deg(f_{ℓ})} ¯f̄_{ℓ}y ⊗ ¯ f̄_{r} ¯f̄′ + (−1)^{deg(f_{r}) deg(x)} ( ¯f̄_{ℓ})x ⊗ ¯ f̄_{r} ¯f̄′)

and the right hand side expands as

(−1)^{deg(f_{r}) deg(x,Σy)} ∑ ( ¯f̄_{ℓ}x ⊗ ¯ f̄_{r} ¯f̄′, (−1)^{deg(f_{ℓ})}Σ ¯f̄_{ℓ}y ⊗ ¯ f̄_{r} ¯f̄′ + ( ¯f̄_{ℓ})x ⊗ ¯ f̄_{r} ¯f̄′);

these two expressions are visibly the same.

Given this left module structure on (𝒱 ˆ⊗𝒱 )_{1}, one can measure the deviation from left equivariance of the diagonal Δ_{𝒱} : 𝒱_{1} → (𝒱 ˆ⊗𝒱 )_{1}. For that, consider the map L̂: 𝒱_{0} ⊗ 𝒱_{1} → (𝒱 ˆ⊗𝒱 )_{1} given by

L̂(f ⊗ x) := Δ_{𝒱} (f x) − f · Δ_{𝒱} (x),

for any f ∈ ℱ_{0} = 𝒱_{0}, x ∈ 𝒱_{1}, where · denotes the left ℱ_{0}-module action defined in (6.9). Since the diagonal Δ_{ℛ} of ℛ^{}is left equivariant, it follows from (6.5) that the image of L̂ lies in the kernel of the map q ˆ⊗q, i. e. in Σ𝒜 ⊗ 𝒜. Moreover if f = ∂v_{1} for some v_{1} ∈ 𝒱_{1}, then one has

Δ_{𝒱} (∂(v_{1})x) = Δ_{𝒱} (v_{1}∂x) = Δ_{𝒱} (v_{1})Δ_{ℱ} (∂x) = Δ_{𝒱} (v_{1})∂ ˆ⊗Δ_{𝒱} (x) = ∂ ˆ⊗Δ_{𝒱} (v_{1})Δ_{𝒱} (x) = Δ_{ℱ} (∂v_{1})Δ_{𝒱} (x),

so that the image of ∂ ⊗ 𝒱_{1} : 𝒱_{1} ⊗ 𝒱_{1} → 𝒱_{0} ⊗ 𝒱_{1} lies in the kernel of L̂. Similarly commutativity of

(6.11)  \xymatrix{𝒱_{1} \ar[r]^{Δ_{𝒱}} \ar[d]_{∂} & (𝒱 ˆ⊗𝒱 )_{1} \ar[d]^{∂ ˆ⊗} \\ 𝒱_{0} \ar[r]^{Δ_{ℱ}} & 𝒱_{0} ⊗ 𝒱_{0}}

implies that 𝒱_{0} ⊗ ker∂ is in the kernel of L̂. It then follows that L factors uniquely through a map

𝒜 ⊗ R_{ℱ} = (𝒱_{0}/ im∂) ⊗ (𝒱_{1}/ ker∂) → ker(q ˆ⊗q) = Σ𝒜 ⊗ 𝒜 .

(6.12) Definition. The map

L_{𝒱} : 𝒜 ⊗ R_{ℱ} → Σ𝒜 ⊗ 𝒜

given by the unique factorization of the map L̂ above is characterized by the deviation of the diagonal Δ_{𝒱} of the Hopf pair algebra 𝒱 from left equivariance. That is, one has

Δ_{𝒱} (f x) = f · Δ_{𝒱} (x) + L_{𝒱} ( ¯f̄ ⊗ ∂x)

for any f ∈ ℱ_{0} = 𝒱_{0}, x ∈ 𝒱_{1} and the action · from (6.9).

Similarly one can measure deviation of Δ_{𝒱} : 𝒱_{1} → (𝒱 ˆ⊗𝒱 )_{1} from cocommutativity by means of the map Ŝ : 𝒱_{1} → (𝒱 ˆ⊗𝒱 )_{1} given by

Ŝ(x) := Δ_{𝒱} (x) − TΔ_{𝒱} (x),

where T : (𝒱 ˆ⊗𝒱 )_{1} → (𝒱 ˆ⊗𝒱 )_{1} is the interchange operator for Alg_{𝟙⊕Σ}^{r} as constructed in (3.13). Then similarly to L̂ above, Ŝ admits a factorization in the following way. First, by commutativity of (6.5) one has

(q ˆ⊗q)TΔ_{𝒱} = T(q ˆ⊗q)Δ_{𝒱} = TΔ_{ℛ} q = Δ_{ℛ} q = (q ˆ⊗q)Δ_{𝒱} ,

since the Alg_{𝟙}^{pair}-comonoid ℛ^{}is cocommutative. Thus the image of Ŝ is contained in ker(q ˆ⊗q) = Σ𝒜 ⊗𝒜. Next, commutativity of (6.11) implies that ker∂ is contained in the kernel of Ŝ. Hence Ŝ factors uniquely as follows

R_{ℱ} = 𝒱_{1}/ ker∂ → ker(q ˆ⊗q) = Σ𝒜 ⊗ 𝒜 .

(6.13) Definition. The map

S_{𝒱} : R_{ℱ} → Σ𝒜 ⊗ 𝒜

given by the unique factorization of the map Ŝ above is characterized by the deviation of the diagonal Δ_{𝒱} of the Hopf pair algebra 𝒱 from cocommutativity. That is, one has

TΔ_{𝒱} (x) = Δ_{𝒱} (x) + S_{𝒱} (∂x)

for any x ∈ 𝒱_{1}.

It is clear from these definitions that L_{𝒱} and S_{𝒱} are well defined maps by the Hopf pair algebra 𝒱 . Below in (12.4) we define the left action operator L : 𝒜 ⊗ R_{ℱ} → Σ𝒜 ⊗ 𝒜 and the symmetry operator S : R_{ℱ} → Σ𝒜 ⊗ 𝒜 with L = 0 and S = 0 if p is odd. For p = 2 these operators are quite intricate but explicitly given. We also will study the dualization of S and L.

The next two results are essentially reformulations of the main results in the book [1].

(6.14) Theorem (Existence). There exists a Hopf pair algebra 𝒱 with L_{𝒱} = L and S_{𝒱} = S.

(6.15) Theorem (Uniqueness). The Hopf pair algebra 𝒱 satisfying L_{𝒱} = L and S_{𝒱} = S is unique up to an isomorphism over the -structure 𝒱 → ℛ^{}and under the kernel 𝒜 ⊕_{}Σ𝒜 𝒱 .

The Hopf pair algebra appearing in these theorems is the algebra of secondary cohomology operations over , denoted by ℬ^{}= (ℬ_{1}^{}→ ℬ_{0}^{}) = ℬ ⊗ . The algebra ℬ has been defined over in [1].

Proof of (6.14). Recall that in [1, 12.1.8] a folding product ˆ⊗ is defined for pair -algebras in such a way that ℬ has a comonoid structure with respect to it, i. e. a secondary Hopf algebra structure. Let

Δ_{1} : ℬ_{1} → (ℬ ˆ⊗ℬ)_{1}

be the corresponding secondary diagonal from [1, (12.2.2)]. It is proved in [1, 14.4] that the left action operator L satisfies

Δ_{1}(bx) = bΔ_{1}(x) + L(q(b) ⊗ (∂x ⊗ 1))

for b ∈ ℬ_{0}, x ∈ ℬ_{1}, ∂x ⊗ 1 ∈ R_{ℬ} ⊗ = ℛ_{1}^{}. Also in [1, 14.5] it is proved that the symmetry operator S satisfies

TΔ_{1}(x) = Δ_{1}(x) + S(∂x ⊗ 1)

for x ∈ ℬ_{1}. Moreover it is proved in [1, 15.3.13] that the secondary Hopf algebra ℬ is determined uniquely up to isomorphism by the maps , L and S.

Consider now the diagram

[figure: Σ𝒜; Σ𝒜; 𝒜 ⊕_{}Σ𝒜; i_{}; q; ℬ_{1} ⊗; ∂⊗1; q=∂⊗1; ℬ_{0} ⊗; 𝒜; 𝒜; R_{ℬ} ⊗; ℱ_{0}; 𝒜 .]

Here the inclusion i_{}: 𝒜 ⊕_{}Σ𝒜 ℬ_{1} ⊗ is given by the inclusion Σ𝒜 ⊂ ℬ_{1} and by the map

𝒜 → ℬ_{1} ⊗

which assigns to an element q(b) ∈ 𝒜, for b ∈ ℬ_{0}, the element [p] · b ⊗ 1. Then it is clear that i_{}is a right 𝒜 -module homomorphism. Moreover it also is a left 𝒜 -module homomorphism since for b ∈ ℬ_{0} the following identity holds in ℬ_{1}:

b · [p] − [p] · b = (b).

Compare [1, A20 in the introduction]. Now one can check that the properties of ℬ established in [1] yield the result.

(6.16) Remark. For elements α, β, γ ∈ 𝒜 with αβ = 0 and βγ = 0 the triple Massey product

⟨α, β, γ⟩ ∈ 𝒜 /(α𝒜 + 𝒜 γ)

is defined. Here the degree of elements in ⟨α, β, γ⟩is deg(α)+deg(β)+deg(γ)−1. We can compute ⟨α, β, γ⟩ by use of the Hopf pair algebra ℬ^{}above as follows. For this we consider the maps

[figure: 𝒜; ℬ_{0} ⊃ R_{ℬ}; q_{ℬ}; q_{R}; R_{ℬ} ⊗ .]

We choose elements ᾱ, β̄, γ̄ ∈ ℬ_{0} which q_{ℬ} carries to α, β, γ respectively. Then we know that the products ᾱβ̄, β̄γ̄ are elements in R_{ℬ} for which we can choose elements x, y ∈ ℬ_{1} ⊗ with

q(x) = q_{R}(ᾱβ̄),
q(y)= q_{R}(β̄γ̄).

Then the bimodule structure of ℬ_{1}⊗ yields the element ᾱy−xγ̄in the kernel Σ𝒜 of q : ℬ_{1}⊗ → R_{ℬ} ⊗ . Now ᾱy − xγ̄ ∈ Σ𝒜 represents ⟨α, β, γ⟩, see [1].

7. The dual of the -relation pair algebra

We next turn to the dualization of the -relation pair algebra of the Steenrod algebra from section 5. For this we just apply the duality functor D to (5.2). There results an exact sequence

[figure: 𝒜_{∗}; ℛ_{}^{0}; d; ℛ_{}^{1}; 𝒜_{∗},]

i. e. the sequence

(7.1)  [figure: 𝒜_{∗}; D(ℛ_{0}^{}); D(∂); D(ℛ_{1}^{}); 𝒜_{∗}; Hom(ℱ_{0}, ); Hom(R_{ℬ} , ).]

In particular, by the dual of (5.3) one has

(7.2) Lemma. The pair ℛ_{}= (d : ℛ_{}^{0} → ℛ_{}^{1}) has a pair coalgebra structure compatible with the standard bicomodule structure of 𝒜_{∗} over itself, so that ℛ_{}yields an object in Coalg_{𝟙}^{pair}, see section 4.

Moreover the dual of (5.4) takes place, i. e. one has

(7.3) Theorem. The pair coalgebra ℛ_{}has a structure of a commutative monoid in the category Coalg_{𝟙}^{pair} with respect to the unfolding product ˇ⊗.

The proof uses the duals of the pair algebras ℛ^{(n)}, n 0, from (5.4). Namely, applying to the short exact sequence

[figure: R_{ℬ}^{(n)}; ℬ_{0}^{⊗n}; q^{⊗n}; 𝒜^{⊗n}]

the functor D = Hom( , ) gives, similarly to (7.2), a pair coalgebra

ℛ_{∗}^{(n)} = ( [figure: 𝒜_{∗}^{⊗n}; ℱ_{∗}^{⊗n}; R_{ℬ}^{(n)}_{∗}; 𝒜_{∗}^{⊗n}] )

such that the following dual of (5.5) holds:

(7.4) Lemma. There is a canonical isomorphism ℛ_{∗}^{(n)} (ℛ_{})^{ˇ⊗n} in Coalg_{𝟙}^{pair}.

Using this lemma one constructs the ˇ⊗-monoid structure on ℛ_{}by the diagram

[figure: ℱ_{∗} ⊗ ℱ_{∗}; Δ_{∗}; ℛ_{}^{0} ⊗ ℛ_{}^{0}; d ˇ⊗; μ; (ℛ_{}ˇ⊗ℛ_{})^{1}; ; μ; R_{ℬ}^{(2)}_{∗}; Δ_{∗}^{}; ℱ_{∗}; ℛ_{}^{0}; d; ℛ_{}^{1}; _{R_{ℬ} _{∗}}]

with Δ^{}as in (5.6).

Moreover the unit of ℛ_{}is given by the dual of (5.7), i. e. by the diagram

[figure: ; 1; ; 0; 1; ; ; 1; 𝒜_{∗}; ℱ_{∗}; _{R_{ℬ} _{∗}}; 𝒜_{∗}]

so that the unit element of R_{ℬ} _{∗} is the map R_{ℬ} → sending the generator p1_{ℬ_{0}} in degree 0 to 1 and all elements in higher degrees to zero.

8. Hopf pair coalgebras

We next turn to the dualization of the notion of a Hopf pair algebra from (6.3), using the dual ℛ_{}of ℛ^{} from the previous section.

(8.1) Definition. A Hopf pair coalgebra 𝒲 (associated to 𝒜_{∗}) is a pair coalgebra d : 𝒲 ^{0} → 𝒲 ^{1} over together with the following commutative diagram in the category of ℱ_{∗}-ℱ_{∗}-bicomodules

[figure: 𝒜_{∗}; ℛ_{}^{0}; ℛ_{}^{1}; i; 𝒜_{∗}; i; 𝒜_{∗}; 𝒲 ^{0}; d; 𝒲 ^{1}; π_{Σ}; _{\binom{π}{π_{Σ}}}; 𝒜_{∗} ⊕_{_{∗}} Σ𝒜_{∗}; Σ𝒜_{∗}; Σ𝒜_{∗}]

with exact rows and columns. The pair morphism i : ℛ_{}→ 𝒲 will be called the -structure of 𝒲 . Moreover 𝒲 must be equipped with a structure of a monoid (m_{𝒲} , 1_{𝒲} ) in Coalg_{𝟙⊕Σ}^{r} such that i is compatible with the Coalg_{𝟙}^{pair}-monoid structure on ℛ_{}from (7.3), i. e. diagrams dual to (6.5) and (6.6)

[figure: (ℛ_{}ˇ⊗ℛ_{})^{1}; m_{ℛ}; i ˇ⊗i; ℛ_{}^{1}; i; (𝒲 ˇ⊗𝒲 )^{1}; m_{𝒲}; 𝒲 ^{1},; ; 1_{ℛ}; ℛ_{}^{1}; i; ⊕ Σ; 1_{𝒲}; 𝒲 ^{1}]

commute.

We next note that the dual of (6.9) holds; more precisely, one has

(8.2) Lemma. For a Hopf pair coalgebra 𝒲 the subspace

(𝒲 ˇ⊗𝒲 )^{1} ⊂ (𝒲 ¯ ¯⊗𝒲 )^{1}

is closed under the left coaction of the coalgebra ℱ_{∗} on (𝒲 ¯ ¯⊗𝒲 )^{1} given by the corestriction of scalars along the multiplication m_{∗} : ℱ_{∗} ⊗ ℱ_{∗} → ℱ_{∗} of the left ℱ_{∗} ⊗ ℱ_{∗} = (𝒲 ¯ ¯⊗𝒲 )^{0}-comodule structure given by the pair coalgebra 𝒲 ¯ ¯⊗𝒲 . In other words, there is a unique map m^{ℓ} : (𝒲 ˇ⊗𝒲 )^{1} → ℱ_{∗} ⊗ (𝒲 ˇ⊗𝒲 )^{1} making the diagram

[figure: (𝒲 ˇ⊗𝒲 )^{1}; m^{ℓ}; ℱ_{∗} ⊗ (𝒲 ˇ⊗𝒲 )^{1}; (𝒲 ¯ ¯⊗𝒲 )^{1}; ℱ_{∗} ⊗ ℱ_{∗} ⊗ (𝒲 ¯ ¯⊗𝒲 )^{1}; m_{∗}⊗1; ℱ_{∗} ⊗ (𝒲 ¯ ¯⊗𝒲 )^{1}]

commute.

Given this left coaction, one can define the dual of the left action operator in (6.12) by measuring deviation of the multiplication (𝒲 ˇ⊗𝒲 )^{1} → 𝒲 ^{1} from being a left comodule homomorphism. For that, one first observes the map Ľ: (𝒲 ˇ⊗𝒲 )^{1} → ℱ_{∗} ⊗ 𝒲 ^{1} given by the difference of two composites in the diagram

\xymatrix{(𝒲 ˇ⊗𝒲 )^{1} \ar[r]^{m^{ℓ}} \ar[d]_{m_{𝒲}} & ℱ_{∗} ⊗ (𝒲 ˇ⊗𝒲 )^{1} \ar[d]^{1⊗m_{𝒲}} \\ 𝒲 ^{1} \ar[r]^{m^{ℓ}} & ℱ_{∗} ⊗ 𝒲 ^{1}.}

Then by the argument dual to that before (6.12) one sees that the map Ľ factors uniquely through coker(i ˇ⊗i) = ((𝒲 ˇ⊗𝒲 )^{1} Σ𝒜_{∗} ⊗ 𝒜_{∗}) and into ker(d) ⊗ im(d) = (𝒜_{∗} ⊗ R_{ℱ} _{∗} 𝒲 ^{0} ⊗ 𝒲 ^{1}) to yield a map Σ𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ R_{ℱ} _{∗}. We thus can make, dually to (6.12), the following

(8.3) Definition. The map

L_{𝒲} : Σ𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ R_{ℱ} _{∗}

given by the unique factorization of the map Ľabove is characterized by the deviation of the multiplication m_{𝒲} of the Hopf pair coalgebra 𝒲 from being a left ℱ_{∗}-comodule homomorphism. That is, for any t ∈ (𝒲 ˇ⊗𝒲 )^{1} one has

(1 ⊗ m_{𝒲} )m^{ℓ}(t) = m^{ℓ}m_{𝒲} (t) + L_{𝒲} (π_{Σ} ˇ⊗π_{Σ})(t).

Next, we define a map S_{𝒲} in a manner dual to (6.13), measuring noncommutativity of the Coalg_{𝟙⊕Σ}^{r} - monoid structure on 𝒲 . For that, we first consider the map Š : (𝒲 ˇ⊗𝒲 )^{1} → 𝒲 ^{1} given by

Š(t) = m_{𝒲} T(t) − m_{𝒲} (t)

for t ∈ (𝒲 ˇ⊗𝒲 )^{1} and then observe that, dually to (6.13), this map factors uniquely through coker(i ˇ⊗i) = ((𝒲 ˇ⊗𝒲 )^{1} Σ𝒜_{∗} ⊗ 𝒜_{∗}) and into im(d) = (R_{ℱ} _{∗} 𝒲 ^{1}) so we have

(8.4) Definition. The map

S_{𝒲} : Σ𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ} _{∗}

given by the unique factorization of the map Š above is characterized by being the graded commutator map with respect to the ˇ⊗-monoid structure on the Hopf pair coalgebra 𝒲 . That is, for any t ∈ (𝒲 ˇ⊗𝒲 )^{1} one has

m_{𝒲} T(t) = m_{𝒲} (t) + S_{𝒲} (π_{Σ} ˇ⊗π_{Σ})(t).

We now dualize the left action operator (12.4) and the symmetry operator (13.1).

(8.5) Definition. The left coaction operator

L_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ R_{ℱ} _{∗}

of degree +1 is the graded dual of the left action operator (12.4).

(8.6) Definition. The cosymmetry operator

S_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ} _{∗}

of degree +1 is the graded dual of the symmetry operator (13.1).

It is clear that the duals of (6.14) and (6.15) hold. Let us state these explicitly.

(8.7) Theorem (Existence). There exists a Hopf pair coalgebra 𝒲 with L_{𝒲} = L_{∗} and S_{𝒲} = S_{∗}.

(8.8) Theorem (Uniqueness). The Hopf pair coalgebra 𝒲 satisfying L_{𝒲} = L_{∗} and S_{𝒲} = S_{∗} is unique up to an isomorphism over 𝒲 𝒜_{∗} ⊕_{_{∗}} Σ𝒜_{∗} and under ℛ_{}𝒲 .

The Hopf pair coalgebra appearing in these theorems will be denoted by ℬ_{}= (ℬ_{}^{0} → ℬ_{}^{1}) = D(ℬ^{}).

Part II. Description of ℬ^{}and ℬ_{}in terms of generators

9. The Milnor dual of the Steenrod algebra

Here we recall the needed facts from [7]. The graded dual of the Hopf algebra 𝒜 is the Milnor Hopf algebra 𝒜_{∗} = Hom(𝒜 , ) = D(𝒜). It is proved in [7] that for odd p as an algebra 𝒜_{∗} is a graded polynomial algebra, i. e. it is isomorphic to a tensor product of an exterior algebra on generators of odd degree and a polynomial algebra on generators of even degree; for p = 2 the algebra 𝒜_{∗} is a polynomial algebra. Moreover, in [7], explicit generators are given in terms of the admissible basis.

First recall that the admissible basis for 𝒜 is given by the following monomials: for odd p they are of the form

M = β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}P^{s_{2}} ⋯ P^{s_{n}}β^{ϵ_{n}}

where ϵ_{k} ∈ {0, 1} and

s_{1} ϵ_{1} + ps_{2}, s_{2} ϵ_{2} + ps_{3}, …, s_{n−1} ϵ_{n−1} + ps_{n}, s_{n} 1.

Then let ξ_{k} ∈ 𝒜_{2(p^{k}−1)} = Hom(𝒜^{2(p^{k}−1)}, ), k 1 and τ_{k} ∈ 𝒜_{2p^{k}−1} = Hom(𝒜^{2p^{k}−1}, ), k 0 be given on this basis by

(9.1)  ξ_{k}(M) = \begin{cases}1, & M = P^{p^{k−1}}P^{p^{k−2}} ⋯ P^{p}P^{1}, \\ 0 & otherwise \end{cases}

and

(9.2)  τ_{k}(M) = \begin{cases}1, & M = P^{p^{k−1}}P^{p^{k−2}} ⋯ P^{p}P^{1}β, \\ 0 & otherwise. \end{cases}

As proved in [7], 𝒜_{∗} is a graded polynomial algebra on these elements, i. e. it is generated by the elements ξ_{k} and τ_{k} with the defining relations

ξ_{i}ξ_{j} = ξ_{j}ξ_{i},
ξ_{i}τ_{j} = τ_{j}ξ_{i},
τ_{i}τ_{j} = −τ_{j}τ_{i}

only.

For p = 2, the admissible basis for 𝒜 is given by the monomials

M = Sq^{s_{1}}Sq^{s_{2}} ⋯ Sq^{s_{n}}

with

s_{1} 2s_{2}, s_{2} 2s_{3}, …, s_{n−1} 2s_{n}, s_{n} 1

and the polynomial generators of 𝒜_{∗} are elements ζ_{k} ∈ 𝒜_{2^{k}−1} = Hom(𝒜^{2^{k}−1}, ) given by

(9.3)  ζ_{k}(M) = \begin{cases}1, & M = Sq^{2^{k−1}}Sq^{2^{k−2}} ⋯ Sq^{2}Sq^{1}, \\ 0 & otherwise. \end{cases}

In terms of these generators, also the coalgebra structure m_{∗} : 𝒜_{∗} → 𝒜_{∗} ⊗ 𝒜_{∗} dual to the multiplication m of 𝒜 is determined in [7]. Namely, for odd p one has

(9.4)  \begin{aligned}m_{∗}(ξ_{k}) & = ξ_{k} ⊗ 1 + ξ_{k−1}^{p} ⊗ ξ_{1} + ξ_{k−2}^{p^{2}} ⊗ ξ_{2} + ⋯ + ξ_{1}^{p^{k−1}} ⊗ ξ_{k−1} + 1 ⊗ ξ_{k}, \\ m_{∗}(τ_{k}) & = ξ_{k} ⊗ τ_{0} + ξ_{k−1}^{p} ⊗ τ_{1} + ξ_{k−2}^{p^{2}} ⊗ τ_{2} + ⋯ + ξ_{1}^{p^{k−1}} ⊗ τ_{k−1} + 1 ⊗ τ_{k} + τ_{k} ⊗ 1. \end{aligned}

For p = 2 one has

(9.5)  m_{∗}(ζ_{k}) = ζ_{k} ⊗ 1 + ζ_{k−1}^{2} ⊗ ζ_{1} + ζ_{k−2}^{4} ⊗ ζ_{2} + ⋯ + ζ_{1}^{2^{k−1}} ⊗ ζ_{k−1} + 1 ⊗ ζ_{k}.

We will need expression of the dual Sq_{∗}^{1} : 𝒜_{∗} → Σ𝒜_{∗} to the map Sq^{1}· : Σ𝒜 → 𝒜 given by multiplication with Sq^{1} from the left.

(9.6) Lemma. The map Sq_{∗}^{1} is equal to \frac{∂}{∂ζ_{1}}. That is, on the monomial basis it is given by

Sq_{∗}^{1}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⋯) = \begin{cases}ζ_{1}^{n_{1}−1}ζ_{2}^{n_{2}} ⋯, & n_{1} ≡ 1 mod 2 \\ 0, & n_{1} ≡ 0 mod 2. \end{cases}

Proof. Note that Sq_{∗}^{1} is a derivation, since Sq^{1}· is a coderivation, i.e. the diagram

[figure: Σ𝒜; Sq^{1}·; Σδ; 𝒜; δ; Σ𝒜 ⊗ 𝒜; _{\binom{Sq^{1}·⊗1}{1⊗Sq^{1}·}}; 𝒜 ⊗𝒜 × 𝒜 ⊗𝒜; +; 𝒜 ⊗ 𝒜]

commutes: indeed for any x ∈ 𝒜 one has

δ(Sq^{1}x) = δ(Sq^{1})δ(x) = (Sq^{1} ⊗ 1 + 1 ⊗ Sq^{1})δ(x) = (Sq^{1} ⊗ 1)δ(x) + (1 ⊗ Sq^{1})δ(x).

On the other hand, on the Milnor generators the derivation Sq_{∗}^{1} acts as follows:

Sq_{∗}^{1}(ζ_{n})(x) = ζ_{n}(Sq^{1}x) = \begin{cases}1, & Sq^{1}x = Sq^{2^{n−1}}Sq^{2^{n−2}} ⋯ Sq^{1}, \\ 0, & Sq^{1}x Sq^{2^{n−1}}Sq^{2^{n−2}} ⋯ Sq^{1}. \end{cases}

It follows that Sq_{∗}^{1}(ζ_{1}) = 1; on the other hand for n > 1 the equation Sq^{1}x = Sq^{2^{n−1}}Sq^{2^{n−2}} ⋯ Sq^{1} has no solutions, since it would imply Sq^{1}Sq^{2^{n−1}}Sq^{2^{n−2}} ⋯ Sq^{1} = Sq^{1}Sq^{1}x = 0, whereas actually

Sq^{1}Sq^{2^{n−1}}Sq^{2^{n−2}} ⋯ Sq^{1} = Sq^{1+2^{n−1}} Sq^{2^{n−2}} ⋯ Sq^{1} 0.

But \frac{∂}{∂ζ_{1}} is the unique derivation sending ζ_{1} to 1 and all other ζ_{n}’s to 0.

We will also need expression of the dual _{∗} of the derivation from (6.1) in terms of the above generators.

(9.7) Lemma. The map _{∗} : Σ𝒜_{∗} → 𝒜_{∗} is equal to the left multiplication by τ_{0} for odd p and by ζ_{1} for p = 2.

Proof. For any linear map ϕ : 𝒜^{n} → the map _{∗}(ϕ) : 𝒜_{n+1} → is the composite of ϕ with : 𝒜_{n+1} → 𝒜_{n}. Thus for p odd one has

(9.8)  _{∗}(ϕ)(β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}P^{s_{2}} ⋯ P^{s_{n}}β^{ϵ_{n}}) = ∑_{ϵ_{k}=1}(−1)^{ϵ_{0}+ϵ_{1}+··· +ϵ_{k−1}}ϕ(β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}⋯ β^{ϵ_{k−1}}P^{s_{k}}P^{s_{k+1}}β^{ϵ_{k+1}} ⋯ P^{s_{n}}β^{ϵ_{n}}).

On the other hand, one has for M as above

(τ_{0}ϕ)(M) = ∑ τ_{0}(M_{ℓ})ϕ(M_{r}) = ∑_{\begin{matrix}M_{ℓ}=cβ \\ 0 c∈ \end{matrix}} cϕ(M_{r}),

if

δ(M) = ∑ M_{ℓ} ⊗ M_{r}.

On the other hand one evidently has

δ(β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}P^{s_{2}} ⋯ P^{s_{n}}β^{ϵ_{n}}) = ∑_{\begin{matrix}0 ι_{0} ϵ_{0} \\ 0 i_{1} s_{1} \\ 0 ι_{1} ϵ_{1} \\ ··· \\ 0 i_{n} s_{n} \\ 0 ι_{n} ϵ_{n} \end{matrix}}(−1)^{∑ _{0 μ<ν n} (ϵ_{μ}−ι_{μ})ι_{ν}}β^{ι_{0}}P^{i_{1}}β^{ι_{1}}⋯ P^{i_{n}}β^{ι_{n}}⊗ β^{ϵ_{0}−ι_{0}}P^{s_{1}−i_{1}} β^{ϵ_{1}−ι_{1}}⋯ P^{s_{n}−i_{n}}β^{ϵ_{n}−ι_{n}}

so that for M = β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}⋯ P^{s_{n}}β^{ϵ_{n}} one has

∑_{\begin{matrix}M_{ℓ}=cβ \\ 0 c∈ \end{matrix}} cϕ(M_{r}) = ∑_{ϵ_{k}=1} ∑_{\begin{matrix}ι_{0}=0 \\ i_{1}=0 \\ ··· \\ i_{k}=0 \\ ι_{k}=1 \\ i_{k+1}=0 \\ ··· \\ i_{n}=0 \\ ι_{n}=0 \end{matrix}}(−1)^{∑ _{0 μ<ν n} (ϵ_{μ}−ι_{μ})ι_{ν}}ϕ(β^{ϵ_{0}−ι_{0}}P^{s_{1}−i_{1}} β^{ϵ_{1}−ι_{1}}⋯ P^{s_{n}−i_{n}}β^{ϵ_{n}−ι_{n}})

= ∑_{ϵ_{k}=1}(−1)^{∑ _{0 μ<k} ϵ_{μ}}ϕ(β^{ϵ_{0}}P^{s_{1}}β^{ϵ_{1}}⋯ P^{s_{k}}P^{s_{k+1}}β^{ϵ_{k+1}} ⋯ P^{s_{n}}β^{ϵ_{n}})

which is the same as (9.8) above.

Similarly for p = 2 the map _{∗}(ϕ) is given by

(9.9)  _{∗}(ϕ)(Sq^{s_{1}} ⋯ Sq^{s_{n}}) = ϕ( (Sq^{s_{1}} ⋯ Sq^{s_{n}})) = ∑_{k=1}^{n} ϕ(Sq^{s_{1}} ⋯ Sq^{s_{k}−1} ⋯ Sq^{s_{n}})

and the map ζ_{1}ϕ is given by

(ζ_{1}ϕ)(M) = ∑ ζ_{1}(M_{ℓ})ϕ(M_{r}) = ∑_{M_{ℓ}=Sq^{1}} ϕ(M_{r}).

On the other hand one has

δ(Sq^{s_{1}} ⋯ Sq^{s_{n}}) = ∑_{\begin{matrix}0 i_{1} s_{1} \\ ··· \\ 0 i_{n} s_{n} \end{matrix}} Sq^{i_{1}} ⋯ Sq^{i_{n}} ⊗ Sq^{s_{1}−i_{1}} ⋯ Sq^{s_{n}−i_{n}},

so that for M = Sq^{s_{1}} ⋯ Sq^{s_{n}} one has

∑_{M_{ℓ}=Sq^{1}} ϕ(M_{r}) = ∑_{k=1}^{n} ∑_{\begin{matrix}i_{1}=0 \\ ··· \\ i_{k−1}=0 \\ i_{k}=1 \\ i_{k+1}=0 \\ ··· \\ i_{n}=0 \end{matrix}} ϕ(Sq^{s_{1}−i_{1}} ⋯ Sq^{s_{n}−i_{n}})

which is equal to (9.9).

It is clear that with respect to the coalgebra structure on 𝒜_{∗} the map _{∗} is a coderivation, i. e. the diagram

[figure: Σ𝒜_{∗}; _{∗}; Σm_{∗}; 𝒜_{∗}; m_{∗}; Σ(𝒜_{∗}⊗𝒜_{∗}); _{\binom{1}{σ}}; Σ𝒜_{∗}⊗𝒜_{∗} ⊕ 𝒜_{∗}⊗Σ𝒜_{∗}; ( _{∗}⊗1,1⊗ _{∗}); 𝒜_{∗} ⊗ 𝒜_{∗}]

is commutative. Here σ is the interchange of Σ as in (3.3). Then using dual of the construction mentioned in (6.2) one may equip the vector space 𝒜_{∗} ⊕ Σ𝒜_{∗} with a structure of an 𝒜_{∗}-𝒜_{∗}-bicomodule, in such a way that one has a short exact sequence of 𝒜_{∗}-𝒜_{∗}-bicomodules

(9.10)  0 → 𝒜_{∗} → 𝒜_{∗} ⊕_{_{∗}} Σ𝒜_{∗} → Σ𝒜_{∗} → 0.

Explicitly, one defines the right coaction of 𝒜_{∗} on 𝒜_{∗} ⊕_{_{∗}} Σ𝒜_{∗} as the direct sum of standard coactions on 𝒜_{∗} and on Σ𝒜_{∗}, whereas the left coaction is given by the composite

𝒜_{∗} ⊕ Σ𝒜_{∗} ⟶^{m_{∗}⊕Σm_{∗}} 𝒜_{∗}⊗𝒜_{∗} ⊕ Σ𝒜_{∗}⊗𝒜_{∗} \underrightarrow{( \begin{matrix}1 & _{∗}⊗1 \\ 0 & σ \end{matrix} )} 𝒜_{∗}⊗𝒜_{∗} ⊕ 𝒜_{∗}⊗Σ𝒜_{∗} 𝒜_{∗} ⊗ (𝒜_{∗} ⊕ Σ𝒜_{∗}).

10. The dual of the tensor algebra ℱ_{0} = T_{}(E_{𝒜} ) for p = 2

We begin by recalling the constructions from [5] relevant to our case.

The Leibniz-Hopf algebra is the free graded associative ring with unit 1 = Z_{0}

(10.1)  𝒵 = T_{}{Z_{1}, Z_{2}, ...}

on generators Z_{n}, one for each degree n 1. Here we use notation as in (5.1). Then 𝒵 is a cocommutative Hopf algebra with respect to the diagonal

Δ(Z_{n}) = ∑_{i=0}^{n} Z_{i} ⊗ Z_{n−i}.

Of course for p = 2 we have 𝒵 ⊗ = ℱ_{0} = T_{}(E_{𝒜} ) by identifying Z_{i} = Sq^{i}, and moreover the diagonal Δ corresponds to Δ^{}⊗ in (5.6). The graded dual of 𝒵 over integers is denoted by ℳ; it is proved in [5] that it is a polynomial algebra. There also a certain set of elements of ℳ is given; it is still a conjecture (first formulated by Ditters) that these elements form a set of polynomial generators for ℳ. If, however, one localizes at any prime p, then there is another set of elements, defined using the so called p-elementary words, which, as proved in [5], is a set of polynomial generators for the localized algebra ℳ. This in particular gives a polynomial generating set for ℱ_{∗} = Hom(ℱ_{0}, _{2}) ℳ /2ℳ. Moreover it turns out that the embedding 𝒜_{∗} ℱ_{∗} given by Hom(𝒜 , _{2}) Hom(ℱ_{0}, _{2}) (dual to the quotient map ℱ_{0} 𝒜) carries the Milnor generators of 𝒜_{∗} to a subset of these generators.

One chooses a basis in ℳ which is dual to the (noncommutative) monomial basis in 𝒵: for any sequence α = (d_{1}, ..., d_{n}) of positive integers, let M_{α} = M_{d_{1},...,d_{n}} be the element of the free abelian group ℳ^{d_{1}+...+d_{n}} = Hom(𝒵^{d_{1}+...+d_{n}}, ) determined by

M_{d_{1},...,d_{n}}(Z_{k_{1}} ⋯ Z_{k_{m}}) = \begin{cases}1, & (k_{1}, ..., k_{m}) = (d_{1}, ..., d_{n}), \\ 0 & otherwise. \end{cases}

Since 𝒵 is a free algebra, dually ℳ is a cofree coalgebra, i. e. the diagonal is given by deconcatenation:

(10.2)  Δ(M_{d_{1},...,d_{n}}) = ∑_{i=0}^{n} M_{d_{1},...,d_{i}}⊗ M_{d_{i+1},...,d_{n}}.

It is noted in [5] (and easy to check) that in this basis the multiplication in ℳ is given by the so called overlapping shuffle product. Rather than defining this rigorously, we will give some examples.

M_{5}M_{2,4,1,9} = M_{5,2,4,1,9}+ M_{7,4,1,9} + M_{2,5,4,1,9}+ M_{2,9,1,9} + M_{2,4,5,1,9}+ M_{2,4,6,9}
+ M_{2,4,1,5,9}+ M_{2,4,1,14} + M_{2,4,1,9,5};
M_{8,5}M_{1,2} = M_{8,5,1,2} + M_{8,6,2} + M_{8,1,5,2} + M_{9,5,2} + M_{8,1,7} + M_{9,7} + M_{1,8,5,2}
+ M_{1,8,7} + M_{1,8,2,5} + M_{9,2,5} + M_{1,2,8,5} + M_{1,10,5} + M_{8,1,2,5}

Thus in general, whereas the ordinary shuffle product of the elements, say, M_{a_{1},a_{2},a_{3}} and M_{b_{1},b_{2},b_{3},b_{4},b_{5}}contains all possible summands like M_{b_{1},a_{1},a_{2},b_{2},b_{3},a_{3},b_{4},b_{5}}, the overlapping shuffle product contains together with each such summand also in addition the summands of the form M_{b_{1}+a_{1},a_{2},b_{2},b_{3},a_{3},b_{4},b_{5}}, M_{b_{1},a_{1},a_{2}+b_{2},b_{3},a_{3},b_{4},b_{5}}, M_{b_{1},a_{1},a_{2},b_{2},b_{3}+a_{3},b_{4},b_{5}}, M_{b_{1},a_{1},a_{2},b_{2},b_{3},a_{3}+b_{4},b_{5}}, M_{b_{1}+a_{1},a_{2}+b_{2},b_{3},a_{3},b_{4},b_{5}} and so on, obtained by replacing an a_{i} and a b_{j} standing one next to other with their sum, in all possible positions.

Note that the algebra of ordinary shuffles is also a polynomial algebra, but over rationals; it is not a polynomial algebra until at least one prime number remains uninverted. On the other hand, over rationals ℳ becomes isomorphic to the algebra of ordinary shuffles.

To define a polynomial generating set for ℳ, we need some definitions. To conform with the admissible basis in the Steenrod algebra, which consists of monomials with decreasing indices, we will reverse the order of indices in the definitions from [5], where the indices go in the increasing order. Thus in our case statements about some M_{d_{1},...,d_{n}}will be equivalent to the corresponding ones in [5] about M_{d_{n},...,d_{1}}.

(10.3) Definitions. The lexicographic order on the basis M_{d_{1},...,d_{n}}of ℳ is defined by declaring M_{d_{1},...,d_{n}}> M_{e_{1},...,e_{m}} if either there is an i with 1 i min(n, m) and d_{i} > e_{i}, d_{n} = e_{m}, d_{n−1} = e_{m−1}, ..., d_{n−i+1} = e_{m−i+1}, d_{n−i} > e_{m−i} or n > m and d_{n−m+1} = e_{1}, d_{n−m+2} = e_{2}, ..., d_{n} = e_{m}.

A basis element M_{d_{1},...,d_{n}} is Lyndon if with respect to this ordering one has M_{d_{1},...,d_{n}} < M_{d_{1},...,d_{i}} for all 1 < i n. For example, M_{3,2,3,2,2} and M_{2,2,1,2,1,2,1}are Lyndon but M_{3,2,2,3,2} and M_{2,1,2,1,2,1} are not.

A basis element M_{d_{1},...,d_{n}}is -elementary if no number > 1 divides all of the d_{i}, i. e. gcd(d_{1}, ..., d_{n}) = 1. The set ESL( ) is the set of elementary basis elements of the form M_{d_{1},...,d_{n},d_{1},...,d_{n},....,d_{1},...,d_{n}}(i. e. d_{1}, ..., d_{n} repeated any number of times), where M_{d_{1},...,d_{n}}is a Lyndon element.

For a prime p, a basis element M_{d_{1},...,d_{n}} is called p-elementary if there is a d_{i} not divisible by p, i. e. p gcd(d_{1}, ..., d_{n}). The set ESL(p) is defined as the set of p-elementary basis elements of the form

M^{d_{1},...,d_{n},d_{1},...,d_{n},...,d_{1},...,d_{n}}_{p^{r} times}

with d_{1}, ..., d_{n} repeated p^{r} times for some r, where M_{d_{1},...,d_{n}}is required to be Lyndon.

For example, M_{15,6,15,6,15,6,15,6} is in ESL(2) but not in ESL( ) or in ESL(p) for any other p, whereas M_{30,6,6} is in ESL(p) for any p 2, 3 but not in ESL(2), not in ESL(3) and not in ESL( ).

One then has

(10.4) Theorem ([5]). The algebra ℳ is a polynomial algebra.

(10.5) Conjecture (Ditters, [5]). The set ESL( ) is the set of polynomial generators for ℳ.

(10.6) Theorem ([5]). For each prime p, the set ESL(p) is a set of polynomial generators for ℳ_{(p)} = ℳ ⊗ _{(p)}, i. e. if one inverts all primes except p.

In particular, it follows that ESL(p) is a set of polynomial generators for ℳ /p^{n} over /p^{n} for all n.

Here are the polynomial generators in low degrees, over and over few first primes. Note that the numbers of generators in each degree are the same (as it should be since all these algebras become isomorphic over ).

1  2  3  4  5
M_{1} M_{1,1} M_{2,1}, M_{1,1,1} M_{3,1}, M_{2,1,1}, M_{1,1,1,1} M_{4,1}, M_{3,2}, M_{3,1,1}, M_{2,2,1}, M_{2,1,1,1}, M_{1,1,1,1,1}

p = 2 M_{1} M_{1,1} M_{3}, M_{2,1} M_{3,1}, M_{2,1,1}, M_{1,1,1,1} M_{5}, M_{4,1}, M_{3,2}, M_{3,1,1}, M_{2,2,1}, M_{2,1,1,1}

p = 3 M_{1} M_{2} M_{2,1}, M_{1,1,1} M_{4}, M_{3,1}, M_{2,1,1}  M_{5}, M_{4,1}, M_{3,2}, M_{3,1,1}, M_{2,2,1}, M_{2,1,1,1}

p = 5 M_{1} M_{2} M_{3}, M_{2,1}  M_{4}, M_{3,1}, M_{2,1,1} M_{4,1}, M_{3,2}, M_{3,1,1}, M_{2,2,1}, M_{2,1,1,1}, M_{1,1,1,1,1}

It is easy to calculate the numbers of polynomial generators in each degree. Let these numbers be m_{1}, m_{2}, · · · . Then the Poincaré series for the algebra ℳ (or 𝒵, or ℱ, or ℱ_{∗}, it does not matter) is

∑_{n} dim(ℳ_{n})t^{n} = (1 − t)^{−m_{1}}(1 − t^{2})^{−m_{2}}(1 − t^{3})^{−m_{3}} ⋯;

on the other hand, we know that it is a tensor coalgebra with one generator in each degree n 1; this implies that dim(ℳ_{n}) = 2^{n−1} for n 1 (and dim(M_{0}) = 1). Thus we have equality of power series

∏_{k=1}^{∞}(1 − t^{k})^{−m_{k}} = 1 + t + 2t^{2} + 4t^{3} + 8t^{4} + ⋯= 1 + t(1 + 2t + (2t)^{2} + (2t)^{3} + ⋯) = 1 + t \frac{1}{1 − 2t} = \frac{1 − t}{1 − 2t}.

Then taking logarithmic derivatives one obtains

∑_{k=1}^{∞} \frac{km_{k}t^{k}}{1 − t^{k}}= \frac{2t}{1 − 2t} − \frac{t}{1 − t} = t + 3t^{2} + 7t^{3} + ⋯ + (2^{n} − 1)t^{n} + · · ·.

It follows that for all n one has

∑_{d|n} dm_{d} = 2^{n} − 1,

which by the Möbius inversion formula gives

m_{n} = \frac{1}{n} ∑_{d|n} μ(d)(2^{\frac{n}{d}} − 1).

The latter expression is well known in the literature on combinatorics; it equals the number of aperiodic bicolored necklaces consisting of n beads, and also the dimension of the nth homogeneous component of the free Lie algebra on two generators. See e. g. [8].

11. The dual of the relation module R_{ℱ}

We now turn to the algebra ℱ_{∗} = Hom(ℱ , _{2}) ℳ /2. By the above, we know that it, as well as ℳ_{(2)}, is a polynomial algebra on the set of generators ESL(2). As an illustration, we will give some expressions of the M-basis elements in terms of sums of overlapping shuffle products of elements from ESL(2). We will give these in ℳ_{(2)} and then their images in ℱ_{∗}.

M_{2} = M_{1}^{2} − 2M_{1,1}

≡ M_{1}^{2} mod 2

M_{1,2} = M_{1}^{3} − M_{3} − M_{2,1} − 2M_{1}M_{1,1}

≡ M_{1}^{3} + M_{3} + M_{2,1} mod 2

M_{1,1,1} = M_{1}M_{1,1} − \frac{1}{3}M_{1}^{3} + \frac{1}{3}M_{3}

≡ M_{1}M_{1,1} + M_{1}^{3} + M_{3} mod 2

M_{4} = \frac{4}{3}M_{1}M_{3} − \frac{1}{3}M_{1}^{4} + 2M_{1,1}^{2} − 4M_{1,1,1,1}

≡ M_{1}^{4} mod 2

M_{2,2} = M_{1,1}^{2} − 2M_{1}^{2}M_{1,1} − \frac{2}{3}M_{1}M_{3} + \frac{2}{3}M_{1}^{4} + 2M_{1,1,1,1}

≡ M_{1,1}^{2} mod 2

M_{1,3} = \frac{1}{3}M_{1}^{4} − \frac{1}{3}M_{1}M_{3} − 2M_{1,1}^{2} − M_{3,1} + 4M_{1,1,1,1}

≡ M_{1}^{4} + M_{1}M_{3} + M_{3,1} mod 2

M_{1,2,1} = M_{1}M_{2,1} − M_{3,1} − M_{1,1}^{2} + 2M_{1}^{2}M_{1,1} + \frac{2}{3}M_{1}M_{3} − \frac{2}{3}M_{1}^{4} − 2M_{1,1,1,1} − 2M_{2,1,1}

≡ M_{1}M_{2,1} + M_{3,1} + M_{1,1}^{2} mod 2

M_{1,1,2} = M_{1,1}^{2} − M_{1}^{2}M_{1,1} − \frac{1}{3}M_{1}M_{3} + \frac{1}{3}M_{1}^{4} − 2M_{1,1,1,1} + M_{3,1} − M_{1}M_{2,1} + M_{2,1,1}

≡ M_{1,1}^{2} + M_{1}^{2}M_{1,1} + M_{1}M_{3} + M_{1}^{4} + M_{3,1} + M_{1}M_{2,1} + M_{2,1,1} mod 2

Moreover it is straightforward to calculate the diagonal in terms of these generators. For example, in ℱ_{∗} one has

Δ(M_{1}) = 1 ⊗ M_{1} + M_{1} ⊗ 1,
Δ(M_{1,1}) = 1 ⊗ M_{1,1} + M_{1} ⊗ M_{1} + M_{1,1} ⊗ 1,
Δ(M_{3}) = 1 ⊗ M_{3} + M_{3} ⊗ 1
Δ(M_{2,1}) = 1 ⊗ M_{2,1} + M_{1}^{2} ⊗ M_{1} + M_{2,1} ⊗ 1
Δ(M_{3,1}) = 1 ⊗ M_{3,1} + M_{3} ⊗ M_{1} + M_{3,1} ⊗ 1
Δ(M_{2,1,1}) = 1 ⊗ M_{2,1,1} + M_{1}^{2} ⊗ M_{1,1} + M_{2,1} ⊗ M_{1} + M_{2,1,1} ⊗ 1
Δ(M_{1,1,1,1}) = 1 ⊗ M_{1,1,1,1} + M_{1} ⊗ M_{1}M_{1,1} + M_{1} ⊗ M_{1}^{3} + M_{1} ⊗ M_{3} + M_{1,1} ⊗ M_{1,1} + M_{1}M_{1,1} ⊗ M_{1}

+ M_{1}^{3} ⊗ M_{1} + M_{3} ⊗ M_{1} + M_{1,1,1,1} ⊗ 1

Δ(M_{4,1}) = 1 ⊗ M_{4,1} + M_{1}^{4} ⊗ M_{1} + M_{4,1} ⊗ 1
Δ(M_{3,2}) = 1 ⊗ M_{3,2} + M_{3} ⊗ M_{1}^{2} + M_{3,2} ⊗ 1
Δ(M_{2,1,1,1}) = 1 ⊗ M_{2,1,1,1} + M_{1}^{2} ⊗ M_{1}M_{1,1} + M_{1}^{2} ⊗ M_{1}^{3} + M_{1}^{2} ⊗ M_{3} + M_{2,1} ⊗ M_{1,1} + M_{2,1,1} ⊗ M_{1}

+ M_{2,1,1,1} ⊗ 1

Δ(M_{5}) = 1 ⊗ M_{5} + M_{5} ⊗ 1
Δ(M_{3,1,1}) = 1 ⊗ M_{3,1,1} + M_{3} ⊗ M_{1,1} + M_{3,1} ⊗ M_{1} + M_{3,1,1} ⊗ 1
Δ(M_{2,2,1}) = 1 ⊗ M_{2,2,1} + M_{1}^{2} ⊗ M_{2,1} + M_{1,1}^{2} ⊗ M_{1} + M_{2,2,1} ⊗ 1.

Also it follows from the results in [5] that one has (11.1) Lemma. For any prime p, in ℳ_{(p)} one has

M_{pd_{1},...,pd_{n}}≡ M_{d_{1},...,d_{n}}^{p} mod p.

To identify the elements to which the Milnor generators ζ_{k} of 𝒜_{∗} go under the isomorphism ℱ_{∗} ℳ /2, we first identify 𝒜_{∗} with the graded dual of 𝒜; then ζ_{k} corresponds to a linear form 𝒜_{2^{k}−1} → given by (9.3).

(11.2) Proposition. Under the embedding 𝒜_{∗} ℳ /2, the Milnor generator ζ_{k} maps to the generator M_{2^{k−1},2^{k−2},...,2,1}. In particular, this generator is in ESL(2), i. e. is one of the polynomial generators of ℱ_{∗}.

Note how this together with (10.2) and (11.1) implies the Milnor formula (9.5) for the diagonal in 𝒜_{∗}. Identifying ζ_{k} with its image in ℳ /2 by (11.2), one obtains

m_{∗}(ζ_{k}) = Δ(M_{2^{k−1},2^{k−2},...,2,1}) = ∑_{i=0}^{k} M_{2^{k−1},2^{k−2},...,2^{i}} ⊗ M_{2^{i−1},...,2,1} = ∑_{i=0}^{k} M_{2^{k−1−i} ,2^{k−2−i} ,...,2,1}^{2^{i}}⊗ M_{2^{i−1},...,2,1}

= ∑_{i=0}^{k} ζ_{k−i}^{2^{i}} ⊗ ζ_{i}.

Thus the set {ζ_{1}, ζ_{2}, ...} of polynomial generators for 𝒜_{∗} can be identified with the subset

Q = {M_{1}, M_{2,1}, M_{4,2,1}, M_{8,4,2,1}, ...}

of the set of polynomial generators ESL(2) for ℳ /2 ℱ_{∗}. This in particular gives an explicit basis for R_{ℱ} _{∗}: it is in one-to-one correspondence with those monomials in the generators M_{d_{1},...,d_{n}} from ESL(2) not all of whose variables belong to Q. For example, in first few low dimensions this basis contains the following monomials:

M_{1,1}, M_{1}M_{1,1}, M_{3},

M_{1}^{2}M_{1,1}, M_{1}M_{3}, M_{1,1}^{2}, M_{3,1}, M_{2,1,1}, M_{1,1,1,1},

M_{1}^{3}M_{1,1}, M_{1}^{2}M_{3}, M_{1}M_{1,1}^{2}, M_{1}M_{3,1}, M_{1}M_{2,1,1}, M_{1}M_{1,1,1,1}, M_{1,1}M_{3}, M_{1,1}M_{2,1}, M_{5}, M_{4,1}, M_{3,2}, M_{3,1,1}, M_{2,2,1}, M_{2,1,1,1}.

We next note that obviously the embedding 𝒜_{∗} ℱ_{∗} identifies ℱ_{∗} with a polynomial algebra over 𝒜_{∗}, namely one has a canonical isomorphism

(11.3)  ℱ_{∗} 𝒜_{∗}[ESL(2) \ Q].

In particular, as an 𝒜_{∗}-module ℱ_{∗} is free on the generating set ^{(ESL(2)\Q)} (= the free commutative monoid on ESL(2) \ Q). Then obviously the quotient module R_{ℱ} _{∗} is a free 𝒜_{∗}-module with the generating set (ESL(2)\Q) \ {1}.

We will need the dual ℱ_{∗}^{2} of the subspace ℱ_{0}^{2} ⊂ ℱ_{0} spanned by the monomials of length 2 in the generators Sq^{i}. Observe that ℱ_{0}^{2} is a subcoalgebra of ℱ_{0}, so that dually ℱ_{∗} ℱ_{∗}^{2} is a quotient algebra. We have

(11.4) Proposition. The algebra ℱ_{∗}^{2} is a quotient of the polynomial algebra on three generators M_{1}, M_{1,1}, M_{2,1} by a single relation

M_{1}M_{1,1}M_{2,1} + M_{1,1}^{3} + M_{2,1}^{2} = 0.

Proof. First of all, it is straightforward to calculate in ℱ_{∗} the sum of the overlapping shuffle products

M_{1}M_{1,1}M_{2,1} + M_{1,1}^{3} + M_{2,1}^{2} = M_{1,4,1} + M_{2,2,2} + M_{2,3,1} + M_{3,1,2} + M_{3,2,1} + M_{2,1,2,1} + M_{3,1,1,1} + M_{1,1,3,1} + M_{1,2,1,1,1}+ M_{1,2,1,2} + M_{1,1,1,2,1}

so that indeed this gives zero in ℱ_{∗}^{2}. Let

X = [x_{1}, x_{2}, x_{3}]/(x_{1}x_{2}x_{3} + x_{2}^{3} + x_{3}^{2})

be the graded algebra with deg(x_{i}) = i, i = 1, 2, 3, so that there is a homomorphism of algebras f : X → ℱ_{∗}^{2} sending x_{1} ↦ M_{1}, x_{2} ↦ M_{1,1}, x_{3} ↦ M_{2,1}. It is straightforward to calculate the Hilbert function of X, i. e. the formal power series

∑_{n} dim(X_{n})t^{n};

it is equal to

\frac{1 − t^{6}}{(1 − t)(1 − t^{2})(1 − t^{3})}.

On the other hand ℱ_{∗}^{2} is dual to ℱ_{0}^{2} and it is straightforward also to calculate dimensions of homogeneous components of this space. One then simply checks that these dimensions coincide for X and for ℱ_{∗}^{2}. Thus it suffices to show that f is surjective, i. e. that ℱ_{∗}^{2} is generated by (the images of) M_{1}, M_{1,1} and M_{2,1}.

We will show by induction on degree that every M_{n} and M_{i,j} can be obtained as a polynomial in these three elements. In degree 1, M_{1} is the only nonzero element. In degree 2, besides M_{1,1} we have M_{2} which is equal to M_{1}^{2} by (11.1). In degree 3, we have

M_{1}M_{1,1} = M_{1,2} + M_{2,1} + M_{1,1,1} ≡ M_{1,2} + M_{2,1} mod ℱ_{∗}^{>2}

and

M_{1}^{3} = M_{3} + M_{1,2} + M_{2,1},

so that in ℱ_{∗}^{2} we may solve

M_{1,2} ≡ M_{1}M_{1,1} + M_{2,1}

and

M_{3} ≡ M_{1}^{3} + M_{1}M_{1,1}.

Given now any degree n > 3, we can obtain any element M_{i,j} with i > 1, j > 1, i + j = n from elements of lower degree since

M_{i,j} ≡ M_{1,1}M_{i−1,j−1} .

Next we also can obtain the element M_{n−1,1} from

M_{n−1,1} + M_{2,n−2} ≡ M_{2,1}M_{n−3}.

Then we can obtain M_{1,n−1} from

M_{1,n−1} + M_{n−1,1} ≡ M_{1,1}M_{n−2},

and finally we can obtain M_{n} from

M_{n} + M_{1,n−1} + M_{n−1,1} ≡ M_{1}M_{n−1}.

Let us also identify the dual of the product map

ℱ_{0}^{1} ⊗ ℱ_{0}^{1} → ℱ_{0}^{2}

in terms of the above generators. By dualizing it is clear that this dual is the unique factorization in the diagram

\xymatrix{ℱ_{∗} \ar[r]^{m_{∗}} \ar@{->>}[d] & ℱ_{∗} ⊗ ℱ_{∗} \ar@{->>}[d] \\ ℱ_{∗}^{2} \ar@{-->}[r] & ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}.}

In particular, it is an algebra homomorphism. Moreover the algebra ℱ_{∗}^{1} may be identified with the polynomial algebra on a single generator M_{1} = ζ_{1}, with the quotient map ℱ_{∗} → ℱ_{∗}^{1} given by sending M_{1} to itself and all other polynomial generators from ESL(2) to zero. From this it is straightforward to identify the map ℱ_{∗}^{2} → ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} with the algebra homomorphism

[x_{1}, x_{2}, x_{3}]/(x_{1}x_{2}x_{3} + x_{2}^{3} + x_{3}^{2}) → [y_{1}, z_{1}]

given by

(11.5)  \begin{matrix}x_{1} & ↦ y_{1} + z_{1} \\ x_{2} & ↦ y_{1}z_{1} \\ x_{3} & ↦ y_{1}^{2}z_{1}. \end{matrix}

Let us identify in these terms the map ℱ_{∗}^{2} R_{ℱ} _{∗}^{2}. One clearly has

R_{ℱ}^{2} = R_{ℱ} ∩ ℱ_{0}^{2}

in ℱ_{0}, so that dually one has that the diagram

\xymatrix{ℱ_{∗} \ar@{->>}[r] \ar@{->>}[d] & R_{ℱ} _{∗} \ar@{->>}[d] \\ ℱ_{∗}^{2} \ar@{->>}[r] & _{R_{ℱ}^{2}_{∗}}}

is pushout. Thus R_{ℱ}^{2}_{∗} is isomorphic to the quotient of ℱ_{∗}^{2} by the image of the composite 𝒜_{∗} ℱ_{∗} ℱ_{∗}^{2}. That image is clearly the subalgebra generated by M_{1} and M_{2,1}.

We can alternatively describe R_{ℱ}^{2}_{∗} in terms of linear forms on R_{ℱ}^{2} ⊂ ℱ_{0}^{2}. It is clear that the latter subspace is spanned by all Adem relations [n, m], n < 2m. The map π : ℱ_{∗}^{2} R_{ℱ}^{2}_{∗} assigns to a linear form on ℱ_{0}^{2} its restriction to R_{ℱ}^{2}. One then clearly has

(11.6)  π(M_{1}^{k}) = π(M_{2,1}^{k}) = 0

for all k 0; moreover π(M_{1,1}) is dual to [1, 1] in the basis given by the elements [n, m], i. e. M_{1,1}([1, 1]) = 1 and M_{1,1}([n, m]) = 0 for all other n, m. Moreover for x, y ∈ ℱ_{∗}^{2} we have

(11.7)  (xy)([n, m]) = ∑ x([n, m]_{ℓ})y([n, m]_{r})

in the Sweedler notation

Δ([n, m]) = ∑ [n, m]_{ℓ} ⊗ [n, m]_{r}.

For example, we have

Δ([1, 2]) = (1 + T)(1 ⊗ [1, 2] + Sq^{1} ⊗ [1, 1])

which implies that M_{1}M_{1,1} is dual to [1, 2] in this basis, i. e. (M_{1}M_{1,1})[1, 2] = 1 and (M_{1}M_{1,1})[n, m] = 0 for all other n, m. Similarly

Δ([1, 3]) = (1 + T)(1 ⊗ [1, 3] + Sq^{1} ⊗ [1, 2] + Sq^{2} ⊗ [1, 1])

and

Δ([2, 2]) = (1 + T)(1 ⊗ [2, 2] + Sq^{1} ⊗ [1, 2] + Sq^{2} ⊗ [1, 1]) + [1, 1] ⊗ [1, 1]

imply that M_{1,1}^{2} is dual to [2, 2] whereas (M_{1}^{2}M_{1,1})[1, 3] = (M_{1}^{2}M_{1,1})[2, 2] = 1, so that dual to [1, 3] is M_{1}^{2}M_{1,1} + M_{1,1}^{2}.

We will also need a description of the dual R̄_{∗}of R̄= R_{ℱ} /(R_{ℱ} · R_{ℱ} ). For this first note that similarly to the above ℱ_{∗} ⊗ ℱ_{∗} is a free 𝒜_{∗} ⊗ 𝒜_{∗}-module on ^{(ESL(2)\Q)}× ^{(ESL(2)\Q)} and R_{ℱ} _{∗} ⊗ R_{ℱ} _{∗} is a free 𝒜_{∗} ⊗ 𝒜_{∗}- module on ( ^{(ESL(2)\Q)} \ {1}) × ( ^{(ESL(2)\Q)} \ {1}). Moreover the diagonal Δ_{ℱ} : ℱ_{∗} → ℱ_{∗} ⊗ ℱ_{∗} and its factorization Δ_{R}: R_{ℱ} _{∗} → R_{ℱ} _{∗} ⊗ R_{ℱ} _{∗} through the quotient maps ℱ_{∗} R_{ℱ} _{∗}, ℱ_{∗} ⊗ ℱ_{∗} R_{ℱ} _{∗} ⊗ R_{ℱ} _{∗} are obviously both equivariant with respect to the diagonal δ : 𝒜_{∗} → 𝒜_{∗} ⊗ 𝒜_{∗}, i. e. one has

(11.8)  \begin{aligned}Δ_{ℱ} (af) & = δ(a)Δ_{ℱ} (f), \\ Δ_{R}(ar) & = δ(a)Δ_{R}(r) \end{aligned}

for any a ∈ 𝒜_{∗}, f ∈ ℱ_{∗}, r ∈ R_{ℱ} _{∗}.

12. The left action operator L and its dual

We next recall constructions of certain maps L and S from [1, 14.4,14.5] of the same kind as the operators in (6.12) and (6.13) respectively. For that, we first introduce the following notation:

(12.1)  R̄:= R_{ℱ} /(R_{ℱ} · R_{ℱ} ),

with the quotient map R_{ℱ} R̄denoted by r ↦ r̄. There is a well-defined 𝒜 -𝒜 -bimodule structure on R̄ given by

¯f̄ r̄= \overline{f r}, r̄ ¯f̄ = \overline{rf}

for f ∈ ℱ_{0}, r ∈ R_{ℱ} . As we show below R̄is free both as a left and as a right 𝒜 -module (but not as a bimodule). A basis for R̄as a right 𝒜 -module can be found using the set PAR ⊂ R_{ℱ} of preadmissible relations as defined in [1, 16.5]. These are the elements of R_{ℱ} of the form

Sq^{n_{1}} ⋯ Sq^{n_{k}}[n, m]

where [n, m], n < 2m, is an Adem relation, the monomial Sq^{n_{1}} ⋯ Sq^{n_{k}} is admissible (i. e. n_{1} 2n_{2}, n_{2} 2n_{3}, ..., n_{k−1} 2n_{k}), and moreover n_{k} 2n. It is then proved in [1, 16.5.2] that PAR is a basis of R_{ℱ} as a free right ℱ_{0}-module. Using this, one also has

(12.2) Lemma. As a right 𝒜 -module, R̄ is free. Moreover, the images ρ̄ of the preadmissible relations ρ ∈ PAR under the quotient map R_{ℱ} R̄form a basis of this free right 𝒜 -module.

Proof. One has

R_{ℱ} /(R_{ℱ} · R_{ℱ} ) (⨁_{ρ∈PAR}ρℱ_{0}) /((⨁_{ρ∈PAR}ρℱ_{0}) · R_{ℱ} )

= (⨁_{ρ∈PAR}ρℱ_{0}) /(⨁_{ρ∈PAR}ρ (ℱ_{0} · R_{ℱ} ))

= (⨁_{ρ∈PAR}ρℱ_{0}) /(⨁_{ρ∈PAR}ρR_{ℱ} )

⨁_{ρ∈PAR} ρ̄(ℱ_{0}/R_{ℱ} )

= ⨁_{ρ∈PAR} ρ̄𝒜;

moreover the above equalities/isomorphisms are easily seen to be compatible with the right 𝒜 -module structure.

In particular we see that every element of R_{ℱ} can be written in a unique way in the form

(12.3)  ρ^{(2)} + ∑_{i} α_{i}[n_{i}, m_{i}]β_{i}

with ρ^{(2)} ∈ R_{ℱ} · R_{ℱ} , α_{i}[n_{i}, m_{i}] ∈ PAR and β_{i} an admissible monomial.

(12.4) Definition. The left action operator

L : 𝒜 ⊗ R_{ℱ} → 𝒜 ⊗ 𝒜

of degree −1 is defined as follows. For odd p let L be the zero map. For p = 2, let first the additive map L_{ℱ} : ℱ_{0}^{2} → 𝒜 ⊗ 𝒜 be given by the formula

L_{ℱ} (Sq^{n}Sq^{m}) = ∑_{\begin{matrix}n_{1}+n_{2}=n \\ m_{1}+m_{2}=m \\ m_{1}, n_{2} odd \end{matrix}}Sq^{n_{1}}Sq^{m_{1}} ⊗ Sq^{n_{2}}Sq^{m_{2}}

(n, m 0; remember that Sq^{0} = 1). Equivalently, using the algebra structure on 𝒜 ⊗ 𝒜 one may write

L_{ℱ} (Sq^{n}Sq^{m}) = (1 ⊗ Sq^{1})δ(Sq^{n−1})(Sq^{1} ⊗ 1)δ(Sq^{m−1}).

Restricting this map to R_{ℱ}^{2} ⊂ ℱ_{0}^{2} gives a map L_{R} : R_{ℱ}^{2} → 𝒜 ⊗ 𝒜. It is thus an additive map given on the Adem relations [n, m], for 0 < n < 2m, by

L_{R}[n, m] = L_{ℱ} (Sq^{n}Sq^{m}) + ∑_{k=max\{0,n−m+1\}}^{min\{n/2,m−1\}}\binom{m − k − 1}{n − 2k}L_{ℱ} (Sq^{n+m−k}Sq^{k}).

Next we define the map

L̄: 𝒜 ⊗ R̄→ 𝒜 ⊗ 𝒜

as the right 𝒜 -module homomorphism which satisfies

(12.5)  L̄(a ⊗ \overline{α[n, m]}) = δ( (a) ¯ᾱ)L_{R}[n, m]

with α[n, m] ∈ PAR; by (12.2) such a homomorphism exists and is unique.

Finally, L̄ yields a unique linear map L : 𝒜 ⊗ R_{ℱ} → 𝒜 ⊗ 𝒜 by composing L̄ with the quotient map 𝒜 ⊗ R_{ℱ} 𝒜 ⊗ R̄. Thus one has

L (𝒜 ⊗ (R_{ℱ} · R_{ℱ} )) = 0.

The map L is the left action operator in [1, 14.4] where the following lemma is proved (see [1, 14.4.3]):

(12.6) Lemma. The map L̄ satisfies the equalities

L̄(a ⊗ [n, m]) = (a)L_{R}[n, m]
L̄(a ⊗ br) = L̄(ab ⊗ r) + δ(a)L̄(b ⊗ r)
L̄(a ⊗ rb) = L̄(a ⊗ r)δ(b)

for any a, b ∈ 𝒜, r ∈ R̄.

We observe that L can be alternatively constructed as follows. Let

L̃: R̄→ 𝒜 ⊗ 𝒜

be the map given by

L̃(r̄) = L̄(Sq^{1} ⊗ r̄).

Then one has

(12.7) Proposition. For any a ∈ 𝒜, r ∈ R_{ℱ} one has

L(a ⊗ r) = δ( (a))L̃(r̄);

moreover L̃is a homomorphism of 𝒜 -𝒜 -bimodules, hence uniquely determined by its values on the Adem relations, which are

L̃([n, m]) = L_{R}[n, m].

Proof. For any a ∈ 𝒜, α[n, m] ∈ PAR and β admissible we have

L(a ⊗ α[n, m]β) = L̄(a ⊗ \overline{α[n, m]})¯β̄
= δ( (a) ¯ᾱ)L_{R}[n, m]δ¯β̄
= δ (a)δ ¯ᾱL_{R}[n, m]δ¯β̄
= δ (a)δ( (Sq^{1}) ¯ᾱ)L_{R}[n, m]δ¯β̄
= δ (a)L(Sq^{1} ⊗ α[n, m]β)
= δ (a)L̃(α[n, m]β).

Then using (12.3) we see that the same identity holds for L(a ⊗ r) with any r ∈ R_{ℱ} .

Next for any a ∈ 𝒜, r ∈ R_{ℱ} we have by (12.6) and Sq^{1} = Sq^{0} = 1,

L̃(ar̄) = L̄(Sq^{1} ⊗ ar̄)
= L̄(Sq^{1}a ⊗ r̄) + δ(Sq^{1})L̄(a ⊗ r̄)
= δ( (Sq^{1}a))L̃(r̄) + δ(Sq^{1} (a))L̃(r̄)
= δ( (Sq^{1}a) + Sq^{1} (a))L̃(r̄)
= δ(a)L̃(r̄).

Thus L̃ is a left 𝒜 -module homomorphism. It is also clearly a right 𝒜 -module homomorphism since L̄ is. Finally by (12.5) we have

L_{R}[n, m] = δ( (Sq^{1}))L_{R}[n, m] = L̄(Sq^{1} ⊗ \overline{[n, m]}) = L̃([n, m]).

Explicit calculation of the left coaction operator L_{∗} is as follows. For odd p it is the zero map, and for p = 2 we first define the additive map L_{R}_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ} _{∗}^{2}. It is dual to the composite map R_{ℱ}^{2} → 𝒜 ⊗ 𝒜 in the diagram

(12.8)  [figure: R_{ℱ}^{2}; pull; R_{ℱ}; ℱ_{0}^{1} ⊗ ℱ_{0}^{1}; Δ⊗Δ; m; ℱ_{0}^{2}; L_{ℱ}; ℱ_{0}; ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1}; 1⊗Φ⊗Φ⊗1; ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1}; 1⊗T⊗1; ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ⊗ ℱ_{0}^{1}; m⊗m; ℱ_{0}^{2} ⊗ ℱ_{0}^{2}; ℱ_{0} ⊗ ℱ_{0}; 𝒜 ⊗ 𝒜]

where Φ is restriction ℱ_{0}^{1} → ℱ_{0}^{1} of the map ℱ_{0} → ℱ_{0} given by

Φ(x) = Sq^{1} (x),

so that one has

Φ(Sq^{n}) = \begin{cases}Sq^{n}, & n ≡ 1 mod 2 \\ 0, & n ≡ 0 mod 2. \end{cases}

Indeed by (12.4) we have

L_{ℱ} (Sq^{n}Sq^{m}) = (1 ⊗ Sq^{1})Δ(Sq^{n−1})(Sq^{1} ⊗ 1)Δ(Sq^{m−1}) = (1 ⊗ Sq^{1})Δ (Sq^{n})(Sq^{1} ⊗ 1)Δ (Sq^{m});

on the other hand we saw in (6.7) that one has

Δ = ( ⊗ 1)Δ = (1 ⊗ )Δ,

so that we can write

L_{ℱ} (Sq^{n}Sq^{m}) = (1 ⊗ Sq^{1} )Δ(Sq^{n})(Sq^{1} ⊗ 1)Δ(Sq^{m}) = (1 ⊗ Φ)Δ(Sq^{n})(Φ ⊗ 1)Δ(Sq^{m}).

So the map dual of Φ is the map Φ_{∗} : [ζ_{1}] → [ζ_{1}] given by factorization through 𝒜_{∗} [ζ_{1}] of the map Φ_{∗} : 𝒜_{∗} → 𝒜_{∗} given on the monomial basis by

Φ_{∗}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⋯) = \begin{cases}ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⋯, & n_{1} ≡ 1 mod 2 \\ 0, & n_{1} ≡ 0 mod 2. \end{cases}

Equivalently, by (9.6) and (9.7), Φ_{∗} = _{∗}Sq_{∗}^{1} is the map ζ_{1} \frac{∂}{∂ζ_{1}}.

Thus the map L_{R}_{∗} is the composite 𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ}^{2}_{∗} in the diagram

(12.9)  [figure: 𝒜_{∗} ⊗ 𝒜_{∗}; ℱ_{∗} ⊗ ℱ_{∗}; ℱ_{∗}^{2} ⊗ ℱ_{∗}^{2}; m_{∗}⊗m_{∗}; _{L_{ℱ} _{∗}}; ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}; 1⊗T⊗1; ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}; 1⊗Φ_{∗}⊗Φ_{∗}⊗1; ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}; Δ_{∗}⊗Δ_{∗}; ℱ_{∗}; push; ℱ_{∗}^{2}; m_{∗}; ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}; _{R_{ℱ} _{∗}}; _{R_{ℱ}^{2}_{∗}}]

Now by (12.7) we know that L̃ is a bimodule homomorphism, and moreover R̄is generated by R_{ℱ}^{2} R̄^{2} ⊂ R̄as an 𝒜 -𝒜 -bimodule, so knowledge of L_{R} (actually already of L_{ℱ} whose restriction it is) determines L̃ and, by (12.7), also L. Dually, one can reconstruct L̃_{∗} and then L_{∗} from L_{ℱ} _{∗} via the diagram

[figure: 𝒜_{∗} ⊗ 𝒜_{∗}; L̃_{∗}; bicoaction; R̄_{∗}; bicoaction; 𝒜_{∗} ⊗ R̄_{∗}⊗ 𝒜_{∗}; 𝒜_{∗} ⊗ (𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ 𝒜_{∗}; 1⊗L_{R}⊗1; 𝒜_{∗} ⊗ R_{ℱ}^{2}_{∗} ⊗ 𝒜_{∗}; 𝒜_{∗} ⊗ R_{ℱ} _{∗} ⊗ 𝒜_{∗}.]

Here the bicoaction 𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ (𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ 𝒜_{∗} is the composite

[figure: 𝒜_{∗} ⊗ 𝒜_{∗}; m_{∗}^{(2)}⊗m_{∗}^{(2)}; (𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ (𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗}); (142536); (𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ (𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ (𝒜_{∗} ⊗ 𝒜_{∗}); δ_{∗}⊗1⊗1⊗δ_{∗}; 𝒜_{∗} ⊗ (𝒜_{∗} ⊗ 𝒜_{∗}) ⊗ 𝒜_{∗}]

We next note the following

(12.10) Lemma. The map L̃_{∗} is a biderivation, i. e.

L̃_{∗}(x_{1}x_{2}, y)= x_{1}L̃_{∗}(x_{2}, y) + x_{2}L̃_{∗}(x_{1}, y),
L̃_{∗}(x, y_{1}y_{2})= y_{1}L̃_{∗}(x, y_{2}) + y_{2}L̃_{∗}(x, y_{1})

for any x, x_{1}, x_{2}, y, y_{1}, y_{2} ∈ 𝒜_{∗}.

Proof.

It thus follows that L̃_{∗}is fully determined by its values L̃_{∗}(ζ_{n}⊗ζ_{n′}) on the Milnor generators. To calculate the bicoaction on these first note that we have

m_{∗}^{(2)}(ζ_{n}) = (1 ⊗ m_{∗})m_{∗}(ζ_{n}) = ∑_{i+i′=n} ζ_{i}^{2^{i′}} ⊗ m_{∗}(ζ_{i′}) = ∑_{i+j+k=n} ζ_{i}^{2^{j+k}} ⊗ ζ_{j}^{2^{k}} ⊗ ζ_{k},

THE ALGEBRA OF SECONDARY COHOMOLOGY OPERATIONS AND ITS DUAL 37

where as always ζ_{0} = 1. For the coaction on ζ_{n} ⊗ ζ_{n′} this then gives in succession

ζ_{n} ⊗ ζ_{n′} ↦ ∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} ζ_{i}^{2^{j+k}} ⊗ ζ_{j}^{2^{k}} ⊗ ζ_{k} ⊗ ζ_{i′}^{2^{j′+k′}} ⊗ ζ_{j′}^{2^{k′}} ⊗ ζ_{k′}

↦ ∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} ζ_{i}^{2^{j+k}} ⊗ ζ_{i′}^{2^{j′+k′}} ⊗ ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}} ⊗ ζ_{k} ⊗ ζ_{k′}

↦ ∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} ζ_{i}^{2^{j+k}}ζ_{i′}^{2^{j′+k′}} ⊗ ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}} ⊗ ζ_{k}ζ_{k′},

so that for the values of L̃_{∗} we have the equation

ιL̃_{∗}(ζ_{n} ⊗ ζ_{n′}) = ∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} ζ_{i}^{2^{j+k}}ζ_{i′}^{2^{j′+k′}} ⊗ L_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) ⊗ ζ_{k}ζ_{k′}

where ι is the above embedding R̄_{∗} 𝒜_{∗} ⊗ R_{ℱ}^{2}_{∗} ⊗ 𝒜_{∗}. Thus we only have to know the values of L_{ℱ} _{∗} on the elements of the form ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}} for j 0, k 0. Obviously these values are zero for j > 2 or j′ > 2. They are also zero for j = 0 or j′ = 0 since Φ_{∗}(1) = 0. There thus remain four cases j = j′ = 1, j = j′ = 2, j = 1, j′ = 2, and j = 2, j′ = 1. We then have under L_{ℱ} _{∗}

[figure: ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; m_{∗}⊗m_{∗}; (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{2^{k}} ⊗ (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{2^{k′}} =; ζ_{1}^{2^{k}} ⊗ 1 ⊗ ζ_{1}^{2^{k′}} ⊗ 1 + ζ_{1}^{2^{k}} ⊗ 1 ⊗ 1 ⊗ ζ_{1}^{2^{k′}} + 1 ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}} ⊗ 1 + 1 ⊗ ζ_{1}^{2^{k}} ⊗ 1 ⊗ ζ_{1}^{2^{k′}}; 1⊗T⊗1; ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}} ⊗ 1 ⊗ 1 + ζ_{1}^{2^{k}} ⊗ 1 ⊗ 1 ⊗ ζ_{1}^{2^{k′}} + 1 ⊗ ζ_{1}^{2^{k′}} ⊗ ζ_{1}^{2^{k}} ⊗ 1 + 1 ⊗ 1 ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; 1⊗Φ_{∗}⊗Φ_{∗}⊗1; 0 + 0 + 1 ⊗ Φ_{∗}ζ_{1}^{2^{k′}} ⊗ Φ_{∗}ζ_{1}^{2^{k}} ⊗ 1 + 0; Δ_{∗}⊗Δ_{∗}; Φ_{∗}ζ_{1}^{2^{k′}} ⊗ Φ_{∗}ζ_{1}^{2^{k}}.]

We thus have

L_{ℱ} _{∗}(ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}) = \begin{cases}M_{1,1}, & k = k′ = 0 \\ 0 & otherwise. \end{cases}

We next take j = j′ = 2; then

[figure: ζ_{2}^{2^{k}} ⊗ ζ_{2}^{2^{k′}}; m_{∗}⊗m_{∗}; (ζ_{1}^{2} ⊗ ζ_{1})^{2^{k}} ⊗ (ζ_{1}^{2} ⊗ ζ_{1})^{2^{k′}} = ζ_{1}^{2^{k+1}} ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′+1}} ⊗ ζ_{1}^{2^{k′}}; 1⊗T⊗1; ζ_{1}^{2^{k+1}} ⊗ ζ_{1}^{2^{k′+1}} ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; 1⊗Φ_{∗}⊗Φ_{∗}⊗1; ζ_{1}^{2^{k+1}} ⊗ Φ_{∗}ζ_{1}^{2^{k′+1}} ⊗ Φ_{∗}ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}} = 0; Δ_{∗}⊗Δ_{∗}; 0,]

so that

L_{ℱ} _{∗}(ζ_{2}^{2^{k}} ⊗ ζ_{2}^{2^{k′}}) = 0

for all k and k′. Next for j = 2, j′ = 1 we have

[figure: ζ_{2}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; m_{∗}⊗m_{∗}; (ζ_{1}^{2} ⊗ ζ_{1})^{2^{k}} ⊗ (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{2^{k′}} = ζ_{1}^{2^{k+1}} ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}} ⊗ 1 + ζ_{1}^{2^{k+1}} ⊗ ζ_{1}^{2^{k}} ⊗ 1 ⊗ ζ_{1}^{2^{k′}}; 1⊗T⊗1; ζ_{1}^{2^{k+1}} ⊗ ζ_{1}^{2^{k′}} ⊗ ζ_{1}^{2^{k}} ⊗ 1 + ζ_{1}^{2^{k+1}} ⊗ 1 ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; 1⊗Φ_{∗}⊗Φ_{∗}⊗1; ζ_{1}^{2^{k+1}} ⊗ Φ_{∗}ζ_{1}^{2^{k′}} ⊗ Φ_{∗}ζ_{1}^{2^{k}} ⊗ 1 + 0; Δ_{∗}⊗Δ_{∗}; ζ_{1}^{2^{k+1}}Φ_{∗}ζ_{1}^{2^{k′}} ⊗ Φ_{∗}ζ_{1}^{2^{k}},]

hence

L_{ℱ} _{∗}(ζ_{2}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}) = \begin{cases}M_{1,1}^{2} + M_{1}M_{2,1}, & k = k′ = 0 \\ 0 & otherwise. \end{cases}

Finally for j = 1, j′ = 2 we get

[figure: ζ_{1}^{2^{k}} ⊗ ζ_{2}^{2^{k′}}; m_{∗}⊗m_{∗}; (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{2^{k}} ⊗ (ζ_{1}^{2} ⊗ ζ_{1})^{2^{k′}} = ζ_{1}^{2^{k}} ⊗ 1 ⊗ ζ_{1}^{2^{k′+1}} ⊗ ζ_{1}^{2^{k′}} + 1 ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′+1}} ⊗ ζ_{1}^{2^{k′}}; 1⊗T⊗1; ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′+1}} ⊗ 1 ⊗ ζ_{1}^{2^{k′}} + 1 ⊗ ζ_{1}^{2^{k′+1}} ⊗ ζ_{1}^{2^{k}} ⊗ ζ_{1}^{2^{k′}}; 1⊗Φ_{∗}⊗Φ_{∗}⊗1; 0 + 0; Δ_{∗}⊗Δ_{∗}; 0,]

so that

L_{ℱ} _{∗}(ζ_{1}^{2^{k}} ⊗ ζ_{2}^{2^{k′}}) = 0

for all k and k′.

To pass to L_{R}_{∗} from these values means just nullifying all monomials which do not contain M_{1,1}; we thus obtain

L_{R}_{∗}(ζ_{1} ⊗ ζ_{1}) = M_{1,1},
L_{R}_{∗}(ζ_{2} ⊗ ζ_{1}) = M_{1,1}^{2},

and L_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) = 0 in all other cases.

From this we easily obtain

(12.11) Proposition. ιL̃_{∗}(ζ_{n} ⊗ ζ_{n′}) = ζ_{n−1}^{2}ζ_{n′−1}^{2} ⊗ M_{1,1} ⊗ 1 + ζ_{n−2}^{4}ζ_{n′−1}^{2} ⊗ M_{1,1}^{2} ⊗ 1 where now ζ_{n−2} = 0 for n = 1 is understood. Solving L̃_{∗}(ζ_{n}, ζ_{n′}) from these equations is then straightforward. In this way we obtain

L̃_{∗}(ζ_{1}, ζ_{1})= M_{1,1}
L̃_{∗}(ζ_{1}, ζ_{2})= M_{2,1,1}
L̃_{∗}(ζ_{2}, ζ_{1})= M_{2,1,1} + M_{1,1}^{2}
L̃_{∗}(ζ_{2}, ζ_{2})= M_{4,1,1} + M_{2,3,1} + M_{2,1,2,1}
L̃_{∗}(ζ_{1}, ζ_{3})= M_{4,2,1,1}
L̃_{∗}(ζ_{3}, ζ_{1})= M_{4,2,1,1} + M_{2,1,1}^{2}
L̃_{∗}(ζ_{2}, ζ_{3})= M_{6,2,1,1} + M_{4,4,1,1} + M_{4,2,3,1} + M_{4,2,1,2,1}+ M_{2,4,2,1,1}
L̃_{∗}(ζ_{3}, ζ_{2})= M_{6,2,1,1} + M_{4,4,1,1} + M_{4,2,3,1} + M_{4,2,1,2,1}+ M_{2,4,2,1,1}
+ M_{5}^{2} + M_{4,1}^{2} + M_{3,2}^{2} + M_{2,1,1,1}^{2} + M_{1}^{2}M_{2,1,1}^{2} + M_{1}^{4}M_{3}^{2}
L̃_{∗}(ζ_{3}, ζ_{3})= M_{8,4,1,1} + M_{8,2,3,1} + M_{8,2,1,2,1}+ M_{4,6,3,1} + M_{4,6,1,2,1}+ M_{4,2,5,2,1}+ M_{4,2,4,3,1}+ M_{4,2,4,1,2,1}
+ M_{4,2,1,4,2,1}
L̃_{∗}(ζ_{1}, ζ_{4})= M_{8,4,2,1,1}
L̃_{∗}(ζ_{4}, ζ_{1})= M_{8,4,2,1,1}+ M_{4,2,1,1}^{2}
L̃_{∗}(ζ_{2}, ζ_{4})= M_{10,4,2,1,1}+ M_{8,6,2,1,1}+ M_{8,4,4,1,1}+ M_{8,4,2,3,1}+ M_{8,4,2,1,2,1}+ M_{8,2,4,2,1,1}+ M_{2,8,4,2,1,1}
L̃_{∗}(ζ_{4}, ζ_{2})= M_{10,4,2,1,1}+ M_{8,6,2,1,1}+ M_{8,4,4,1,1}+ M_{8,4,2,3,1}+ M_{8,4,2,1,2,1}+ M_{8,2,4,2,1,1}+ M_{2,8,4,2,1,1}
+ M_{9}^{2} + M_{7,2}^{2} + M_{5,4}^{2} + M_{6,2,1}^{2} + M_{4,4,1}^{2} + M_{4,3,2}^{2} + M_{4,2,1,1,1}^{2}+ M_{3,4,2}^{2} + M_{2,4,2,1}^{2}
+ M_{1}^{2}M_{4,2,1,1}^{2} + M_{1}^{8}M_{5}^{2} + M_{2,1}^{4}M_{3}^{2}
L̃_{∗}(ζ_{3}, ζ_{4})= M_{12,6,2,1,1}+ M_{12,4,4,1,1}+ M_{12,4,2,3,1}+ M_{12,4,2,1,2,1}+ M_{12,2,4,2,1,1}+ M_{8,8,4,1,1}+ M_{8,8,2,3,1}
+ M_{8,8,2,1,2,1}+ M_{8,4,6,3,1}+ M_{8,4,6,1,2,1}+ M_{8,4,2,5,2,1}+ M_{8,4,2,4,3,1}+ M_{8,4,2,4,1,2,1}+ M_{8,4,2,1,4,2,1}
+ M_{4,10,4,2,1,1}+ M_{4,8,6,2,1,1}+ M_{4,8,4,4,1,1}+ M_{4,8,4,2,3,1}+ M_{4,8,4,2,1,2,1}+ M_{4,8,2,4,2,1,1}+ M_{4,2,8,4,2,1,1},

etc.

Having L̃_{∗} we then can obtain L_{∗} by the dual of (12.7) as

(12.12)  L_{∗}(x, y) = ∑ ζ_{1}x_{ℓ}y_{ℓ′}⊗ L̃_{∗}(x_{r}, y_{r′})

for x, y ∈ 𝒜_{∗}, with

m_{∗}(x) = ∑ x_{ℓ} ⊗ x_{r}, m_{∗}(y) = ∑ y_{ℓ′}⊗ y_{r′}.

13. The symmetry operator S and its dual

(13.1) Definition. The symmetry operator

S : R_{ℱ} → 𝒜 ⊗ 𝒜

of degree −1 is defined as follows. For odd p, let S be the zero map. For p = 2 let the elements S_{n} ∈ 𝒜 ⊗𝒜, n 0, be given by

S_{n} = ∑_{\begin{matrix}n_{1}+n_{2}=n−1 \\ n_{1}, n_{2} odd \end{matrix}} Sq^{n_{1}} ⊗ Sq^{n_{2}} = (Sq^{1} ⊗ Sq^{1})δ(Sq^{n−3}),

i. e.

S_{2k} = 0,
S_{2k+1} = ∑_{0 i<k} Sq^{2i+1} ⊗ Sq^{2(k−i)−1},

k 0. Then let the linear map S_{ℱ} : ℱ_{0}^{2} → 𝒜 ⊗ 𝒜 be given by

S_{ℱ} (Sq^{n}Sq^{m}) = S_{n}δ(Sq^{m}) + δ(Sq^{n})S_{m}+ δ(Sq^{n−1})S_{m+1}

= (Sq^{1} ⊗ Sq^{1})δ(Sq^{n−3}Sq^{m}) + δ(Sq^{n})(Sq^{1} ⊗ Sq^{1})δ(Sq^{m−3}) + δ(Sq^{n−1})(Sq^{1} ⊗ Sq^{1})δ(Sq^{m−2}),

n, m 0. Next define the map S_{R} : R_{ℱ}^{2} → 𝒜 ⊗ 𝒜 by restriction to R_{ℱ}^{2} ⊂ ℱ_{0}^{2}. Thus on the Adem relations this map is given by

(13.2)  S_{R}[n, m] = S_{ℱ} (Sq^{n}Sq^{m}) + ∑_{k=max\{0,n−m+1\}}^{min\{n/2,m−1\}}\binom{m − k − 1}{n − 2k}S_{ℱ} (Sq^{n+m−k}Sq^{k}).

Now let us define the map

S̄ : R̄→ 𝒜 ⊗ 𝒜

as a unique right 𝒜 -module homomorphism satisfying

S̄(\overline{α[n, m]}) = δ(α)S_{R}[n, m] + (1 + T)L̄(α ⊗ \overline{[n, m]})

for α[n, m] ∈ PAR. Then finally this determines a unique linear map S : R_{ℱ} → 𝒜 ⊗ 𝒜 by composing with the quotient map R_{ℱ} R̄.

The map S is the symmetry operator in [1, 14.5.2] where the following lemma is proved.

(13.3) Lemma. The map S̄ satisfies the equations

S̄([n, m])= S_{R}[n, m]
S̄(ar) = δ(a)S̄(r) + (1 + T)L̄(a ⊗ r)
S̄(ra) = S̄(r)δ(a)

for any 0 < n < 2m, a ∈ 𝒜 and r ∈ R̄.

We now turn to the dual S_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ} _{∗} of S (dually to the above, the image of this operator actually lies in R̄_{∗} ⊂ R_{ℱ} _{∗} and so defines the operator S̄_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → R̄_{∗}). Since we know that S_{∗} is a biderivation, it suffices to compute the values S_{∗}(ζ_{n} ⊗ ζ_{n′}). Now dually to the equation

S(a[n, m]b) = δ(a)S_{R}([n, m])δ(b)+(1+T)L(a⊗[n, m]b) = δ(a)S_{R}([n, m])δ(b)+(1+T) (δ (a)L_{R}([n, m])δ(b))

we have

ιS_{∗}(ζ_{n} ⊗ ζ_{n′}) =

∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} (ζ_{i}^{2^{j+k}}ζ_{i′}^{2^{j′+k′}} ⊗ S_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) ⊗ ζ_{k}ζ_{k′}+ ζ_{1}ζ_{i}^{2^{j+k}}ζ_{i′}^{2^{j′+k′}} ⊗ (L_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) + L_{R}_{∗}(ζ_{j′}^{2^{k′}} ⊗ ζ_{j}^{2^{k}})) ⊗ ζ_{k}ζ_{k′})

= ∑_{\begin{matrix}i+j+k=n \\ i′+j′+k′=n′ \end{matrix}} ζ_{i}^{2^{j+k}}ζ_{i′}^{2^{j′+k′}} ⊗ S_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) ⊗ ζ_{k}ζ_{k′}+ ζ_{1}ζ_{n−2}^{4}ζ_{n′−1}^{2} ⊗ M_{1,1}^{2} ⊗ 1 + ζ_{1}ζ_{n−1}^{2}ζ_{n′−2}^{4} ⊗ M_{1,1}^{2} ⊗ 1,

with ζ_{0} = 1 and ζ_{n} = 0 for n < 0, as before.

It thus remains to find the values S_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) — which in turn are images of the corresponding values of S_{ℱ} _{∗} under the map ℱ_{∗} R_{ℱ} _{∗}. To find the latter, let us first define another intermediate operator

S^{1} : ℱ_{0}^{1} → 𝒜 ⊗ 𝒜

by the equation

S^{1}(Sq^{n}) = S_{n+1} = (Sq^{1} ⊗ Sq^{1})δ (Sq^{n}) = ∑_{\begin{matrix}n_{1}+n_{2}=n \\ n_{1}, n_{2} odd \end{matrix}}Sq^{n_{1}} ⊗ Sq^{n_{2}},

so that we have

S_{ℱ} m(Sq^{n} ⊗ Sq^{m}) = S_{ℱ} (Sq^{n}Sq^{m}) = S^{1} (Sq^{n})δ(Sq^{m}) + δ(Sq^{n})S^{1} (Sq^{m}) + δ (Sq^{n})S^{1}(Sq^{m}).

We have the dual operator

S_{∗}^{1} : 𝒜_{∗} ⊗ 𝒜_{∗} → ℱ_{∗}^{1}

THE ALGEBRA OF SECONDARY COHOMOLOGY OPERATIONS AND ITS DUAL 41

such that dual

S_{ℱ} _{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → ℱ_{∗}^{2}

of S_{ℱ} is given by

(13.4)  \begin{matrix}m_{∗}S_{ℱ} _{∗}(x ⊗ y) = \\ ∑ & (ζ_{1}S_{∗}^{1}(x_{ℓ} ⊗ y_{ℓ′}) ⊗ (x_{r}y_{r′})^{1} + (x_{ℓ}y_{ℓ′})^{1} ⊗ ζ_{1}S_{∗}^{1}(x_{r} ⊗ y_{r′}) + (ζ_{1}x_{ℓ}y_{ℓ′})^{1} ⊗ S_{∗}^{1}(x_{r} ⊗ y_{r′})) \end{matrix}

where as before we use the Sweedler notation

m_{∗}(x) = ∑ x_{ℓ} ⊗ x_{r}, m_{∗}(y) = ∑ y_{ℓ′}⊗ y_{r′}

and

( )^{1} : 𝒜_{∗} → ℱ_{∗}^{1}

sends ζ_{1} to M_{1} and all other Milnor generators to 0. Thus we have

m_{∗}S_{ℱ} _{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) =

∑_{\begin{matrix}ℓ+r=j \\ ℓ′+r′=j′ \end{matrix}} ζ_{1}S_{∗}^{1}(ζ_{ℓ}^{2^{r+k}} ⊗ ζ_{ℓ′}^{2^{r′+k′}}) ⊗ (ζ_{r}^{2^{k}}ζ_{r′}^{2^{k′}})^{1} + (ζ_{ℓ}^{2^{r+k}}ζ_{ℓ′}^{2^{r′+k′}})^{1} ⊗ ζ_{1}S_{∗}^{1}(ζ_{r}^{2^{k}} ⊗ ζ_{r′}^{2^{k′}}) + (ζ_{1}ζ_{ℓ}^{2^{r+k}}ζ_{ℓ′}^{2^{r′+k′}})^{1} ⊗ S_{∗}^{1}(ζ_{r}^{2^{k}} ⊗ ζ_{r′}^{2^{k′}})

Now the operator S_{∗}^{1} is obviously given by

(13.5)  S_{∗}^{1}(x ⊗ y) = \begin{cases}xy, & x = ζ_{1}^{n_{1}}, y = ζ_{1}^{n_{2}}, n_{1}, n_{2} odd, \\ 0 & otherwise, \end{cases}

so that S_{ℱ} _{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) = 0 whenever k > 0 or k′ > 0. And among the remaining values S_{ℱ} _{∗}(ζ_{j} ⊗ ζ_{j′}) the only nonzero ones are given by

S_{ℱ} _{∗}(ζ_{1} ⊗ ζ_{1}) = M_{3} + M_{1,2} = M_{1}^{3} + M_{2,1},
S_{ℱ} _{∗}(ζ_{1} ⊗ ζ_{2}) = S_{ℱ} _{∗}(ζ_{2} ⊗ ζ_{1}) = M_{2,3} + M_{3,2} = M_{1}M_{1,1}^{2},
S_{ℱ} _{∗}(ζ_{2} ⊗ ζ_{2}) = M_{5,2} + M_{4,3} = M_{1}M_{2,1}^{2}.

Then further passing to S_{R}_{∗} means, as before, to remove the monomials not containing M_{1,1}, so that the only nonzero values of the form S_{R}_{∗}(ζ_{j}^{2^{k}} ⊗ ζ_{j′}^{2^{k′}}) are

S_{R}_{∗}(ζ_{1} ⊗ ζ_{2}) = S_{R}_{∗}(ζ_{2} ⊗ ζ_{1}) = M_{1}M_{1,1}^{2}.

Hence we obtain (13.6) Proposition. ιS_{∗}(ζ_{n}⊗ζ_{n′}) = ζ_{n−1}^{2}ζ_{n′−2}^{4} ⊗M_{1}M_{1,1}^{2}⊗1+ζ_{n−2}^{4}ζ_{n′−1}^{2} ⊗M_{1}M_{1,1}^{2}⊗1+ζ_{1}ζ_{n−2}^{4}ζ_{n′−1}^{2} ⊗M_{1,1}^{2}⊗1+ζ_{1}ζ_{n−1}^{2}ζ_{n′−2}^{4} ⊗M_{1,1}^{2}⊗1.

As for L̃_{∗} above, we then solve these equations obtaining e. g.

S_{∗}(ζ_{1}, ζ_{1})= 0,
S_{∗}(ζ_{1}, ζ_{2}) = S_{∗}(ζ_{2}, ζ_{1}) = M_{2,2,1} + M_{1}M_{1,1}^{2},
S_{∗}(ζ_{2}, ζ_{2})= 0,
S_{∗}(ζ_{1}, ζ_{3}) = S_{∗}(ζ_{3}, ζ_{1}) = M_{4,2,2,1} + M_{1}M_{2,1,1}^{2},
S_{∗}(ζ_{2}, ζ_{3}) = S_{∗}(ζ_{3}, ζ_{2}) = M_{6,2,2,1} + M_{4,4,2,1} + M_{2,4,2,2,1}
+ M_{1}M_{5}^{2} + M_{1}M_{4,1}^{2} + M_{1}M_{3,2}^{2} + M_{1}M_{2,1,1,1}^{2} + M_{1}^{3}M_{2,1,1}^{2} + M_{1}^{5}M_{3}^{2},
S_{∗}(ζ_{3}, ζ_{3})= 0,
S_{∗}(ζ_{1}, ζ_{4}) = S_{∗}(ζ_{4}, ζ_{1}) = M_{8,4,2,2,1}+ M_{1}M_{4,2,1,1}^{2},
S_{∗}(ζ_{2}, ζ_{4}) = S_{∗}(ζ_{4}, ζ_{2}) = M_{10,4,2,2,1}+ M_{8,6,2,2,1}+ M_{8,4,4,2,1}+ M_{8,2,4,2,2,1}+ M_{2,8,4,2,2,1}
+ M_{1}M_{9}^{2} + M_{1}M_{7,2}^{2} + M_{1}M_{6,2,1}^{2} + M_{1}M_{5,4}^{2} + M_{1}M_{4,4,1}^{2} + M_{1}M_{4,3,2}^{2} + M_{1}M_{4,2,1,1,1}^{2}
+ M_{1}M_{3,4,2}^{2} + M_{1}M_{2,4,2,1}^{2} + M_{1}M_{2,1}^{2}M_{3}^{2} + M_{1}^{3}M_{4,2,1,1}^{2} + M_{1}^{9}M_{5}^{2},

etc.

14. The extended Steenrod algebra

In this section we introduce a singular extension 𝒜̂of the Steenrod algebra 𝒜 which is determined by the symmetry operator S.

(14.1) Definition. A singular extension of a Hopf algebra A is a direct sum diagram

[figure: R; i; Â; p; q; A,; s]

i. e. one has ps = id_{A}, qi = id_{R}and sp+iq = idÂ, such that Âis an algebra with multiplication μ : Â⊗Â→ Â and Âis also a coalgebra with diagonal δ̂: Â→ Â⊗ Â. (Here we do not assume that δ̂is a homomorphism of algebras, or equivalently that μ is a homomorphism of coalgebras, so that in general Â is not a Hopf algebra). In addition p is an algebra homomorphism, and s is a coalgebra homomorphism. Moreover (i, p) must be a singular extension of algebras and (q, s) must be a singular extension of coalgebras. This means that the ideal R = keri of the algebra Âis a square zero ideal, i. e. xy = 0 for any x, y ∈ R, and the coideal R = cokers of the coalgebra Âis a square zero coideal, i. e. the composite

Â⟶^{δ̂} Â⊗ Â ⟶^{q⊗q} R ⊗ R

is zero.

It follows that the Â-Â-bimodule and Â-Â-bicomodule structures on R descend to an A-A-bimodule and A-A-bicomodule structures respectively.

Our basic example of a singular Hopf algebra extension is as follows. We have seen that R̄from (12.1) has an 𝒜 -𝒜 -bimodule structure. Now it also has an 𝒜 -𝒜 -bicomodule structure as follows. On the one hand, there is a diagonal Δ_{R}: R_{ℱ} → R_{ℱ}^{(2)} = ker(q_{ℱ} ⊗ q_{ℱ} ) induced in the commutative diagram

[figure: R_{ℱ}; Δ; ℱ_{0}; q_{ℱ}; Δ; 𝒜; δ; R_{ℱ}^{(2)}; ℱ_{0} ⊗ ℱ_{0}; q_{ℱ} ⊗q_{ℱ}; 𝒜 ⊗ 𝒜]

with short exact rows. Moreover there is a short exact sequence

[figure: R_{ℱ} ⊗ R_{ℱ}; i^{(2)}=\binom{i_{ℱ} ⊗1}{−1⊗i_{ℱ}}; ℱ_{0}⊗R_{ℱ} ⊕ R_{ℱ} ⊗ℱ_{0}; R_{ℱ}^{(2)},]

where i_{ℱ} : R_{ℱ} ↪ ℱ_{0} is the inclusion. Since the composite of the quotient map

ℱ_{0}⊗R_{ℱ} ⊕ R_{ℱ} ⊗ℱ_{0} 𝒜 ⊗R̄⊕ R̄⊗𝒜

with i^{(2)} is obviously zero, we get the induced map

R_{ℱ}^{(2)} → 𝒜 ⊗R̄⊕ R̄⊗𝒜 .

Moreover the diagonal of ℱ_{0} factors through this map as follows

(14.2)  [figure: R̄; _{\binom{Δ_{ℓ}}{Δr}}; R_{ℱ}; i_{ℱ}; Δ; ℱ_{0}; q_{ℱ}; Δ; 𝒜; δ; 𝒜 ⊗R̄⊕ R̄⊗𝒜; R^{(2)}; ℱ_{0} ⊗ ℱ_{0}; q_{ℱ} ⊗q_{ℱ}; 𝒜 ⊗ 𝒜]

giving the left, resp. right coaction Δ_{ℓ}, resp. Δ_{r} of the desired 𝒜 -𝒜 -bicomodule structure on R̄.

Note that the above construction is actually precisely dual to the standard procedure for equipping the kernel of a singular extension with a structure of a bimodule over a base. In particular we could use the dual diagram

(14.3)  [figure: 𝒜 ⊗R̄⊕ R̄⊗𝒜; _{\binom{m_{ℓ}}{mr}}; R^{(2)}; m; ℱ_{0} ⊗ ℱ_{0}; q_{ℱ} ⊗q_{ℱ}; m; 𝒜 ⊗ 𝒜; m; R̄; R_{ℱ}; i_{ℱ}; ℱ_{0}; q_{ℱ}; 𝒜]

to give R̄via m_{ℓ} and m_{r} the structure of 𝒜 -𝒜 -bimodule.

(14.4) Theorem. There is a unique singular extension of Hopf algebras

[figure: Σ^{−1} R̄; i; 𝒜̂; p; q; 𝒜 ,; s]

where 𝒜̂is the split singular extension of algebras, that is, as an algebra

𝒜̂= 𝒜 ⊕ Σ^{−1} R̄

is the semidirect product with multiplication

(a, r)(a′, r′) = (aa′, ar′+ ra′)

and the following conditions are satisfied.

The induced 𝒜 -𝒜 -bimodule and 𝒜 -𝒜 -bicomodule structures on Σ^{−1} R̄are given by the ones indicated in (14.2) above, and the diagonal δ̂of the coalgebra 𝒜̂fits into the commutative diagram

(14.5)  [figure: 𝒜̂; δ̂; 𝒜̂⊗ 𝒜̂; 1+T; Σ^{−1} R̄; S; 𝒜 ⊗ 𝒜; 𝒜̂⊗ 𝒜̂]

where S is the symmetry operator in (13.1).

We will prove this theorem together with the dual statement. Note that clearly the dual of a singular extension of any Hopf algebra A is a singular extension of the dual Hopf algebra A_{∗}. Clearly then the above theorem is equivalent to

(14.6) Theorem. There is a unique singular extension of Hopf algebras

[figure: Σ^{−1} R̄_{∗}; q_{∗}; 𝒜̂_{∗}; s_{∗}; i_{∗}; 𝒜_{∗},; p_{∗}]

where 𝒜̂_{∗} is the split singular extension of coalgebras, that is, as a coalgebra

𝒜̂_{∗} = 𝒜_{∗} ⊕ Σ^{−1} R̄_{∗}

with diagonal

𝒜_{∗} ⊕ Σ^{−1} R̄_{∗} ⟶^{(\begin{matrix}m_{∗} & 0 \\ 0 & m_{ℓ}_{∗} \\ 0 & m_{r}_{∗} \\ 0 & 0 \end{matrix} )} 𝒜_{∗}⊗𝒜_{∗} ⊕ 𝒜_{∗}⊗Σ^{−1} R̄_{∗}⊕ Σ^{−1} R̄_{∗}⊗𝒜_{∗} ⊕ Σ^{−1} R̄_{∗}⊗Σ^{−1} R̄_{∗}

where the diagonal m_{∗} is dual to the multiplication m : 𝒜 ⊗ 𝒜 → 𝒜 and m_{ℓ}_{∗}, m_{r}_{∗} are the 𝒜_{∗}-𝒜_{∗}- bicomodule structure maps dual to the 𝒜 -𝒜 -bimodule structure maps m_{ℓ} : 𝒜 ⊗ Σ^{−1} R̄ → Σ^{−1} R̄, m_{r} : Σ^{−1} R̄⊗ 𝒜 → Σ^{−1} R̄ in (14.3), where the induced 𝒜_{∗}-𝒜_{∗}-bimodule structure on R̄_{∗} is dual to the 𝒜 -𝒜 - bicomodule structure indicated in (14.2) above, and where the multiplication δ̂_{∗} of the algebra 𝒜̂_{∗} satisfies the commutation rule

p_{∗}(y)p_{∗}(x) = p_{∗}(x)p_{∗}(y) + S_{∗}(x ⊗ y)

for any x, y ∈ 𝒜_{∗}, where

S_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → Σ^{−1} R̄_{∗}

is the cosymmetry operator from (8.6).

Proof of (14.4) and (14.6). The diagonal δ̂can be written as follows

𝒜 ⊕ Σ^{−1} R̄ ⟶^{(\begin{matrix}ϕ_{11} & ϕ_{12} \\ ϕ_{21} & ϕ_{22} \\ ϕ_{31} & ϕ_{32} \\ ϕ_{41} & ϕ_{42} \end{matrix} )} 𝒜 ⊗𝒜 ⊕ 𝒜 ⊗Σ^{−1} R̄⊕ Σ^{−1} R̄⊗𝒜 ⊕ Σ^{−1} R̄⊗Σ^{−1} R̄.

Then the condition that s : 𝒜 𝒜 ⊕ Σ^{−1} R̄is a coalgebra homomorphism implies ϕ_{11} = δ and ϕ_{21} = 0, ϕ_{31} = 0, ϕ_{41} = 0. Moreover the condition that the 𝒜 -𝒜 -bicomodule structure induced on Σ^{−1} R̄coincides with the one given in (14.2) implies ϕ_{22} = Δ_{ℓ}, ϕ_{32} = Δ_{r}. Next the condition that (s, q) is a singular extension of coalgebras, i. e. the coideal R̄has zero comultiplication, implies ϕ_{42} = 0. Finally, let us look at the diagram (14.5). The lower composite in this diagram sends (a, r) ∈ 𝒜 ⊕ Σ^{−1} R̄to

(S(r), 0, 0, 0) ∈ 𝒜 ⊗𝒜 ⊕ 𝒜 ⊗Σ^{−1} R̄⊕ Σ^{−1} R̄⊗𝒜 ⊕ Σ^{−1} R̄⊗Σ^{−1} R̄.

The upper composite sends it to

(1 + T)δ̂(a, r) = (1 + T)(δ(a) + ϕ_{12}(r), Δ_{ℓ}(r), Δ_{r}(r), 0)
= ((1 + T)δ(a) + (1 + T)ϕ_{12}(r), Δ_{ℓ}(r) + TΔ_{r}(r), Δ_{r}(r) + TΔ_{ℓ}(r), 0).

Since δ is cocommutative, one has (1 + T)δ = 0. Moreover cocommutativity of Δ : ℱ_{0} → ℱ_{0}⊗ ℱ_{0} implies TΔ_{ℓ} = Δ_{r}, TΔ_{r} = Δ_{ℓ}. Thus commutativity of (14.5) is equivalent to the condition

(14.7)  (1 + T)ϕ_{12} = S : Σ^{−1} R̄→ 𝒜 ⊗ 𝒜 .

Equivalently, passing to the dual we see that the dual map ξ_{∗} = ϕ_{12}_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → Σ^{−1} R̄_{∗}must satisfy

ξ_{∗}(1 + T) = S_{∗}.

Now it is easy to see that ξ_{∗} is in fact the algebra cocycle determining the algebra extension

[figure: R̄_{∗}; q_{∗}; 𝒜̂_{∗}; s_{∗}; 𝒜_{∗},]

that is, in 𝒜̂_{∗} = 𝒜_{∗} ⊕ Σ^{−1} R̄_{∗}one has

(α, β)(α′, β′) = (αα′, αβ′+ βα′+ ξ_{∗}(α ⊗ α′)).

Hence by (14.7) one has

(α, β)(α′, β′) − (α′, β′)(α, β) = (0, S_{∗}(α ⊗ α′)).

Now recall that 𝒜_{∗} is actually a polynomial algebra. Using this fact it has been shown in [1, 16.2] that the algebra structure of any of its singular extensions such as 𝒜̂_{∗} above is completely determined by its commutator map, i. e. by S_{∗}. Thus ϕ_{12}_{∗} and hence the whole ϕ_{ij} matrix is uniquely determined. It is then straightforward to check that indeed this matrix yields a coalgebra structure on 𝒜̂with desired properties.

It follows immediately from (14.6) (and actually this was also deduced during its proof) that one has

(14.8) Corollary. For the cosymmetry operator S_{∗} from (8.6) there exists a map

ξ_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → Σ^{−1} R̄_{∗}

which is a 2-cocycle, i. e. for any x, y, z ∈ 𝒜_{∗} one has

xξ_{∗}(y, z) + ξ_{∗}(x, yz) = zξ_{∗}(x, y) + ξ_{∗}(xy, z)

and such that its symmetrization is equal to S_{∗}, i. e. for any x, y ∈ 𝒜_{∗} one has

ξ_{∗}(x, y) + ξ_{∗}(y, x) = S_{∗}(x, y).

Proof. This follows since any extension

[figure: M; i; A′; p; A]

of a commutative algebra A by a symmetric A-module M is determined by a 2-cocycle c : A ⊗ A → M such that for any x, y ∈ A′one has

xy − yx = i(c(px, py) − c(py, px)),

i. e. the commutator map for A′ is given by the antisymmetrization of c. Of course for p = 2 there is no difference between symmetrization and antisymmetrization.

(14.9) Remark. The above corollary is easily seen to be exactly dual to [1, Theorem 16.1.5].

Using the extended Steenrod algebra we can next compute the deviation of the cocycle ξ_{∗} from being an 𝒜_{∗}-comodule homomorphism. Namely, let

∇_{ξ} _{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ Σ^{−1} R̄_{∗}

be the difference between the upper and lower composites in the diagram

(14.10)  [figure: 𝒜_{∗} ⊗ 𝒜_{∗}; ξ_{∗}; coaction; 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗}; 1⊗ξ_{∗}; Σ^{−1} R̄_{∗}; coaction; 𝒜_{∗} ⊗ Σ^{−1} R̄_{∗}.]

Thus on elements we have

∇_{ξ} _{∗}(x, y) = ∑ ξ_{∗}(x, y)_{𝒜} ⊗ ξ_{∗}(x, y)_{R}− ∑ x_{ℓ}y_{ℓ′}⊗ ξ_{∗}(x_{r}, y_{r′}),

where again the Sweedler notation is used,

m_{∗}(x) = ∑ x_{ℓ} ⊗ x_{r}

for the diagonal

m_{∗} : 𝒜_{∗} → 𝒜_{∗} ⊗ 𝒜_{∗}

and

a_{∗}(x) = ∑ x_{𝒜} ⊗ x_{C}

for the coaction

a_{∗} : C → 𝒜_{∗} ⊗ C

of a left 𝒜_{∗}-comodule C.

Let us also denote by ∇_{S} _{∗} the similar operator but with S_{∗} in place of ξ_{∗}. That is, we define

∇_{S} _{∗}(x, y) = ∑ S_{∗}(x, y)_{𝒜} ⊗ S_{∗}(x, y)_{R}− ∑ x_{ℓ}y_{ℓ′}⊗ S_{∗}(x_{r}, y_{r′}).

We then obviously have

(14.11)  ∇_{ξ} _{∗}(x, y) + ∇_{ξ} _{∗}(y, x) = ∇_{S} _{∗}(x, y)

for any x, y ∈ 𝒜_{∗}.

(14.12) Lemma. The map ∇_{ξ} _{∗} above is a 2-cocycle, i. e. for any x, y, z ∈ 𝒜_{∗} one has

m_{∗}(x)∇_{ξ} _{∗}(y, z) + ∇_{ξ} _{∗}(x, yz) = ∇_{ξ} _{∗}(x, y)m_{∗}(z) + ∇_{ξ} _{∗}(xy, z).

Proof. First note that the diagram

\xymatrix{𝒜_{∗} ⊗ R̄_{∗} \ar[r]^{m_{∗}⊗coaction} \ar[dd]_{action} & 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ R̄_{∗} \ar@{-->}[dr]^{1⊗T⊗1} \\  &  & 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ R̄_{∗} \ar[d]^{δ_{∗}⊗action} \\ R̄_{∗} \ar[rr]^{coaction} &  & 𝒜_{∗} ⊗ R̄_{∗}}

commutes — this follows from the fact that the action and coaction of 𝒜_{∗} on R̄_{∗} are induced from the multiplication and comultiplication in ℱ_{∗} which is a Hopf algebra.

We thus conclude that the coaction map

R̄_{∗}→ 𝒜_{∗} ⊗ R̄_{∗}

is a homomorphism of 𝒜_{∗}-modules, so that its compliste with the cocycle ξ_{∗} is a cocycle. It thus remains to show that the composite

𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ 𝒜_{∗} → 𝒜_{∗} ⊗ R̄_{∗}

in the diagram (14.10) is also a cocycle. Let us denote this composite by ϕ.

Now observe that the Hopf algebra diagram for 𝒜_{∗} expressing interchange of the multiplication and diagonal can be written on elements as follows:

∑ (xy)_{ℓ} ⊗ (xy)_{r} = ∑ x_{ℓ}y_{ℓ′}⊗ x_{r}y_{r′}.

Using this identity we then have for any x, y, z ∈ 𝒜_{∗}

m_{∗}(x)ϕ(y, z)= ∑ x_{ℓ}y_{ℓ′}z_{ℓ′′}⊗ x_{r}ξ_{∗}(y_{r′}, z_{r′′});
ϕ(x, yz)= ∑ x_{ℓ}(yz)_{ℓ′}⊗ ξ_{∗}(x_{r}, (yz)_{r′}) = (δ_{∗} ⊗ ξ_{∗}) (∑ x_{ℓ} ⊗ (yz)_{ℓ′}⊗ x_{r} ⊗ (yz)_{r′})
= (δ_{∗} ⊗ ξ_{∗}) (∑ x_{ℓ} ⊗ y_{ℓ′}z_{ℓ′′}⊗ x_{r} ⊗ y_{r′}z_{r′′}) = ∑ x_{ℓ}y_{ℓ′}z_{ℓ′′}⊗ ξ_{∗}(x_{r}, y_{r′}z_{r′′});
ϕ(xy, z)= ∑ (xy)_{ℓ}z_{ℓ′}⊗ ξ_{∗}((xy)_{r}, z_{r′}) = (δ_{∗} ⊗ ξ_{∗}) (∑ (xy)_{ℓ} ⊗ z_{ℓ′}⊗ (xy)_{r} ⊗ z_{r′})
= (δ_{∗} ⊗ ξ_{∗}) (∑ x_{ℓ}y_{ℓ′}⊗ z_{ℓ′′}⊗ x_{r}y_{r′} ⊗ z_{r′′}) = ∑ x_{ℓ}y_{ℓ′}z_{ℓ′′}⊗ ξ_{∗}(x_{r}y′_{r}, z_{r′′});
ϕ(x, y)m_{∗}(z)= ∑ x_{ℓ}y_{ℓ′}z_{ℓ′′}⊗ ξ_{∗}(x_{r}, y_{r′})z_{r′′}.

These indentities readily imply that ϕ is a cocycle as required.

We next use the fact the cocycle ∇_{ξ} _{∗} is defined on a polynomial algebra and hence can be expressed by its values on generators and by its (anti)symmetrization ∇_{S} _{∗}. Indeed the proof of [1, 16.2.3] works in this generality, i. e. one has

(14.13) Proposition. Let P = k[ζ_{1}, ζ_{2}, ...] be a polynomial algebra over a commutative ring k, let M be a P-module, let

γ : P ⊗ P → M

be a Hochschild 2-cocycle, i. e. one has

xγ(y, z) − γ(xy, z) + γ(x, yz) − zγ(x, y) = 0

for all x, y, z ∈ P, and let σ be the antisymmetrization of γ, i. e.

σ(x, y) = γ(x, y) − γ(y, x).

Then, up to coboundaries, γ can be recovered from σ, i. e. there is a cocycle γ_{σ} cohomologous to γ which depends only on σ.

Proof. To γ corresponds a singular extension of k-algebras

[figure: M; i; E; p; P]

whose isomorphism class uniquely determines the cohomology class of γ. Let us choose for each polynomial generator ζ_{n} ∈ P an element s(ζ_{n}) ∈ E with ps(ζ_{n}) = ζ_{n}. Furthermore let us choose an ordering on the polynomial generators of P, ζ_{1} < ζ_{2} < ...; these data determine uniquely a k-linear section of p, by the formula

s(ζ_{n_{1}}ζ_{n_{2}} ⋯) = s(ζ_{n_{1}})s(ζ_{n_{2}}) ⋯

for any finite sequence n_{1} n_{2} ⋯of positive integers. Then we can use s to construct the cocycle γ_{σ} cohomologous to γ determined by

s(xy) = s(x)s(y) + iγ_{σ}(x, y).

But if x and y are monomials, then s(xy) and s(x)s(y) differ only by the order of terms, so that iγ_{σ}(x, y) is contained in the ideal generated by commutators

γ_{σ}(ζ_{i}, ζ_{j}) = s(ζ_{i})s(ζ_{j}) − s(ζ_{j})s(ζ_{i}) = σ(ζ_{i}, ζ_{j})

for i > j. So in fact one can express each γ_{σ}(x, y) by a linear combination of elements of M of the form zσ(ζ_{i}, ζ_{j}) for z ∈ P.

(14.14) Remark. Obviously the above proof actually contains an algorithm for expressing the cocycle γ_{σ} in terms of σ. For x = ζ_{n_{1}}ζ_{n_{2}} ⋯ ζ_{n_{k}} and y = ζ_{m_{1}}ζ_{m_{2}}⋯ ζ_{m_{l}}, with n_{1} n_{2} ⋯ n_{k}, m_{1} m_{2} ⋯ m_{l}, either one has n_{k} m_{1}, in which case γ_{σ}(x, y) = 0 since s(x)s(y) = s(xy), or one has n_{k} > m_{1}, in which case one can write

s(x)s(y) = s(ζ_{n_{1}}) ⋯ s(ζ_{n_{k−1}})s(ζ_{m_{1}})s(ζ_{n_{k}})s(ζ_{m_{2}}) ⋯ s(ζ_{m_{l}}) + ζ_{n_{1}} ⋯ ζ_{n_{k−1}}ζ_{m_{2}}⋯ ζ_{m_{l}}σ(ζ_{m_{1}}, ζ_{n_{k}}).

Applying the same procedure again several times one finally arrives at s(xy)+(a sum of elements of the form zσ(ζ_{i}, ζ_{j})). In fact it is easy to see that one has

γ_{σ}(ζ_{n_{1}}ζ_{n_{2}} ⋯ ζ_{n_{k}}, ζ_{m_{1}}ζ_{m_{2}}⋯ ζ_{m_{l}}) = ∑_{n_{i}>m_{j}} ζ_{n_{1}} ⋯ ζ_{n_{i−1}}ζ_{n_{i+1}} ⋯ ζ_{n_{k}}ζ_{m_{1}}⋯ ζ_{m_{j−1}}ζ_{m_{j+1}} ⋯ ζ_{m_{l}}σ(ζ_{m_{j}}, ζ_{n_{i}}).

In the characteristic p > 0 case further obvious simplifications occur. In particular we can choose the cocycle ξ_{∗} in (14.8) in such a way that the formula

(14.15)  \begin{matrix}ξ_{∗}(ζ_{1}^{d_{1}}ζ_{2}^{d_{2}} ⋯, ζ_{1}^{e_{1}}ζ_{2}^{e_{2}} ⋯) = \\ ∑_{\begin{matrix}i<j \\ e_{i}, d_{j} odd \end{matrix}}ζ_{1}^{d_{1}+e_{1}} ⋯ ζ_{i−1}^{d_{i−1}+e_{i−1}} ζ_{i}^{d_{i}+e_{i}−1} ζ_{i+1}^{d_{i+1}+e_{i+1}} ⋯ ζ_{j−1}^{d_{j−1} +e_{j−1}} ζ_{j}^{d_{j}+e_{j}−1} ζ_{j+1}^{d_{j+1} +e_{j+1}} ⋯ S_{∗}(ζ_{i}, ζ_{j}) \end{matrix}

holds, and similarly for ξ_{∗} and S_{∗} replaced by ∇_{ξ} _{∗} and ∇_{S} _{∗} respectively:

(14.16)  \begin{matrix}∇_{ξ} _{∗}(ζ_{1}^{d_{1}}ζ_{2}^{d_{2}} ⋯, ζ_{1}^{e_{1}}ζ_{2}^{e_{2}} ⋯) = \\ ∑_{\begin{matrix}i<j \\ e_{i}, d_{j} odd \end{matrix}}ζ_{1}^{d_{1}+e_{1}} ⋯ ζ_{i−1}^{d_{i−1}+e_{i−1}} ζ_{i}^{d_{i}+e_{i}−1} ζ_{i+1}^{d_{i+1}+e_{i+1}} ⋯ ζ_{j−1}^{d_{j−1} +e_{j−1}} ζ_{j}^{d_{j}+e_{j}−1} ζ_{j+1}^{d_{j+1} +e_{j+1}} ⋯ ∇_{S} _{∗}(ζ_{i}, ζ_{j}). \end{matrix}

The operator ∇_{S} _{∗} is readily computable. It is a symmetric biderivation, with ∇_{S} _{∗}(x, x) = 0 for all x, thus uniquely determined by its values of the form ∇_{S} _{∗}(ζ_{n}, ζ_{m}) for n < m, which are expressed easily from the corresponding values of S_{∗}. For example, one has

∇_{S} _{∗}(ζ_{1}, ζ_{2})= ζ_{1} ⊗ M_{1,1}^{2},
∇_{S} _{∗}(ζ_{1}, ζ_{3})= ζ_{1}^{5} ⊗ M_{1,1}^{2} + ζ_{1} ⊗ M_{2,1,1}^{2},
∇_{S} _{∗}(ζ_{2}, ζ_{3})= (ζ_{1}^{7} + ζ_{1}ζ_{2}^{2})⊗ M_{1,1}^{2} + ζ_{1}^{3} ⊗ M_{2,1,1}^{2} + ζ_{1} ⊗ (M_{1}^{2}M_{3} + M_{1}M_{2,1,1} + M_{5} + M_{4,1} + M_{3,2} + M_{2,1,1,1}) ^{2} ,
∇_{S} _{∗}(ζ_{1}, ζ_{4})= ζ_{1}ζ_{2}^{4} ⊗ M_{1,1}^{2} + ζ_{1}^{9} ⊗ M_{2,1,1}^{2} + ζ_{1} ⊗ M_{4,2,1,1}^{2},
∇_{S} _{∗}(ζ_{2}, ζ_{4})= (ζ_{1}^{3}ζ_{2}^{4} + ζ_{1}ζ_{3}^{2})⊗ M_{1,1}^{2} + ζ_{1}^{11} ⊗ M_{2,1,1}^{2}
+ ζ_{1}^{9} ⊗ (M_{1}^{2}M_{3} + M_{1}M_{2,1,1} + M_{5} + M_{4,1} + M_{3,2} + M_{2,1,1,1}) ^{2} + ζ_{1}^{3} ⊗ M_{4,2,1,1}^{2}
+ ζ_{1} ⊗ (M_{1}^{4}M_{5} + M_{2,1}^{2}M_{3} + M_{1}^{2}M_{4,2,1,1} + M_{9} + M_{7,2} + M_{6,2,1} + M_{5,4} + M_{3,4,2} + M_{4,3,2} + M_{4,4,1}
+M_{2,4,2,1} + M_{4,2,1,1,1})^{2} ,

etc.

We will next use certain elements defined in [6, Theorem 3.3] to derive more explicit expressions for ξ_{∗}, hence for S_{∗}, ∇_{S} _{∗} and ∇_{ξ} _{∗}. We recall that Kristensen defines

A[a, b] = (Sq^{1} ⊗ Sq^{0,1})δ (Sq^{a−3}Sq^{b−2} + Sq^{a−2}Sq^{b−3} + ∑_{j} \binom{b − 1 − j}{a − 2j}(Sq^{a+b−j−3} Sq^{j−2} + Sq^{a+b−j−2} Sq^{j−3} )) ,

for natural numbers a, b. Obviously one has

A[a, b] = (Sq^{1} ⊗ Sq^{0,1})δk([a, b]),

where k is the operator determined by

k(xy) = ( (x) (y))

for x, y ∈ ℱ_{0}^{1}. We then interpret A[a, b] as an -linear operator of the form

K : ℱ_{0}^{1} ⊗ ℱ_{0}^{1} → 𝒜 ⊗ 𝒜

given by

K(x ⊗ y) = (Sq^{1} ⊗ Sq^{0,1})δ ( (x) (y))

which is factored through ℱ_{0}^{1} ⊗ ℱ_{0}^{1} ℱ_{0}^{2} and then restricted to R_{ℱ}^{2} ℱ_{0}^{2}. We then can dualize K to get

(14.17) Definition. We define an -linear operator

K_{∗} : 𝒜_{∗} ⊗ 𝒜_{∗} → R_{ℱ}^{2}_{∗}

as composite with the quotient map ℱ_{∗}^{2} R_{ℱ}^{2}_{∗} of the dual of K above (whose image lies in that of m_{∗} : ℱ_{∗}^{2} ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}.

Thus explicitly, K_{∗} is the composite

𝒜_{∗} ⊗ 𝒜_{∗} ⟶^{Sq^{1}·_{∗}⊗Sq^{0,1}·_{∗}} 𝒜_{∗} ⊗ 𝒜_{∗} ⟶^{δ_{∗}} 𝒜_{∗} ⟶^{ζ_{1}} 𝒜_{∗} ℱ_{∗} ⟶^{m_{∗}} ℱ_{∗} ⊗ ℱ_{∗} ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} ⟶^{M_{1}^{2}⊗M_{1}^{2}} ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1}

landing in ℱ_{∗}^{2} ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} and precomposed with ℱ_{∗}^{2} R_{ℱ}^{2}_{∗}. Or on elements,

K_{∗}(x ⊗ y) = (M_{1}^{2} ⊗ M_{1}^{2}) (m_{∗}(ζ_{1} \frac{∂x}{∂ζ_{1}} \frac{∂y}{∂ζ_{2}}))^{1} = m_{∗}(M_{1,1}^{2}M_{1} \frac{∂x}{∂ζ_{1}} \frac{∂y}{∂ζ_{2}})^{1}.

One thus has

(14.18)  K_{∗}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⋯ ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}⋯) = \begin{cases}M_{1}^{n_{1}+m_{1}}M_{2,1}^{n_{2}+m_{2}−1}M_{1,1}^{2}, & n_{1}, m_{2} odd, n_{i} = m_{i} = 0 for i > 2, \\ 0 & otherwise. \end{cases}

We have

(14.19) Proposition. Symmetrization of the operator K_{∗} dual to the operator S_{R} in (13.2), i. e. is given by precomposing S_{ℱ} _{∗} given in (13.4) with the restriction map ℱ_{∗}^{2} R_{ℱ} _{∗}^{2}.

Proof. From the above formula (14.18), for monomials x = ζ_{1}^{n_{1}}ζ_{2}^{n_{2}}ζ_{3}^{n_{3}} ⋯and y = ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}ζ_{3}^{m_{3}}⋯we have

K_{∗}(x ⊗ y) + K_{∗}(y ⊗ x) = \begin{cases}M_{1}^{n_{1}+m_{1}}M_{2,1}^{n_{2}+m_{2}−1}M_{1,1}^{2}, & n_{1}m_{2} + m_{1}n_{2} odd and n_{i} = m_{i} = 0 for i > 2, \\ 0 & otherwise. \end{cases}

On the other hand, using the explicit expression (13.4) and the expression for the operator S_{∗}^{1} in (13.5) we can write

m_{∗}S_{ℱ} _{∗}(x ⊗ y) = ∑_{\begin{matrix}x_{ℓ}=ζ_{1}^{2n−1} \\ y_{ℓ′}=ζ_{1}^{2n′−1} \end{matrix}} ζ_{1}^{2(n+n′)−1} ⊗ (x_{r}y_{r′})^{1} + (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1}) ∑_{\begin{matrix}x_{r}=ζ_{1}^{2n−1} \\ y_{r′}=ζ_{1}^{2n′−1} \end{matrix}}(x_{ℓ}y_{ℓ′})^{1} ⊗ ζ_{1}^{2(n+n′−1)}.

From the expression (9.5) for the Milnor diagonal we thus see that for monomials x = ζ_{1}^{n_{1}}ζ_{2}^{n_{2}}ζ_{3}^{n_{3}} ⋯and y = ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}ζ_{3}^{m_{3}}⋯one has S_{ℱ} _{∗}(x ⊗ y) = 0 unless n_{i} = m_{i} = 0 for i > 2, whereas in the remaining cases one has

m_{∗}S_{ℱ} _{∗}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}) = ∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \\ i, j odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})+1} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}}

+ (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1}) ∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \\ n_{1} − i + n_{2}, m_{1} − j + m_{2} odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}} .

Let us now turn back to the symmetrization of K_{∗}. We compute its image under the map m_{∗}; by (11.5) it sends M_{1} to ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1}, M_{1,1} to ζ_{1} ⊗ ζ_{1} and M_{2,1} to ζ_{1}^{2} ⊗ ζ_{1}. Thus the nonzero values of this image are, for n_{1}m_{2} + m_{1}n_{2} odd,

m_{∗}K_{∗}(1 + T)(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}) = (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{n_{1}+m_{1}}(ζ_{1}^{2} ⊗ ζ_{1})^{n_{2}+m_{2}−1}(ζ_{1}^{2} ⊗ ζ_{1}^{2}).

Then expanding (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{n_{1}+m_{1}} = (ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{n_{1}}(ζ_{1} ⊗ 1 + 1 ⊗ ζ_{1})^{m_{1}} via binomials we obtain

m_{∗}K_{∗}(1 + T)(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}) = ∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}+1} .

It follows that nonzero values of the difference m_{∗}(S_{ℱ} _{∗} − K_{∗}(1 + T)) on monomials in Milnor generators are equal to

∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \\ i, j odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})+1} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}}

+ ∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \\ n_{1} − i + n_{2}, m_{1} − j + m_{2} odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})+1} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}}

+ ∑_{\begin{matrix}0 i n_{1} \\ 0 j m_{1} \\ n_{1} − i + n_{2}, m_{1} − j + m_{2} even \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{j}ζ_{1}^{i+j+2(n_{2}+m_{2})} ⊗ ζ_{1}^{n_{1}+m_{1}−i−j+n_{2}+m_{2}+1}

for n_{1}m_{2} + m_{1}n_{2} odd and m_{∗}S_{ℱ} _{∗}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}) for n_{1}m_{2} + m_{1}n_{2} even.

The first expression can be rewritten as

(ζ_{1}^{2} ⊗ ζ_{1})^{n_{2}+m_{2}} ∑_{k} ζ_{1}^{k+1} ⊗ ζ_{1}^{n_{1}+m_{1}−k}

( ∑_{\begin{matrix}0 i n_{1} \\ 0 k−i m_{1} \\ i, k − i odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k − i} + ∑_{\begin{matrix}0 i n_{1} \\ 0 k−i m_{1} \\ n_{1} − i + n_{2}, m_{1} − k + i + m_{2} odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k − i} + ∑_{\begin{matrix}0 i n_{1} \\ 0 k+1−i m_{1} \\ n_{1} − i + n_{2}, m_{1} − k − 1 + i + m_{2} even \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k + 1 − i})

and in the second case we may write

m_{∗}S_{ℱ} _{∗}(ζ_{1}^{n_{1}}ζ_{2}^{n_{2}} ⊗ ζ_{1}^{m_{1}}ζ_{2}^{m_{2}}) = (ζ_{1}^{2} ⊗ ζ_{1})^{n_{2}+m_{2}} ∑_{k} ζ_{1}^{k+1} ⊗ ζ_{1}^{n_{1}+m_{1}−k}

( ∑_{\begin{matrix}0 i n_{1} \\ 0 k−i m_{1} \\ i, k − i odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k − i} + ∑_{\begin{matrix}0 i n_{1} \\ 0 k−i m_{1} \\ n_{1} − i + n_{2}, m_{1} − k + i + m_{2} odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k − i} + ∑_{\begin{matrix}0 i n_{1} \\ 0 k+1−i m_{1} \\ n_{1} − i + n_{2}, m_{1} − k − 1 + i + m_{2} odd \end{matrix}} \binom{n_{1}}{i}\binom{m_{1}}{k + 1 − i}) .

One then shows that these expressions lie in the subalgebra of ℱ_{∗}^{1} ⊗ ℱ_{∗}^{1} generated by ζ_{1}^{2} ⊗ ζ_{1} and ζ_{1}⊗ 1 + 1 ⊗ ζ_{1}, without involvement of ζ_{1}⊗ ζ_{1}. This means that the image of the difference S_{ℱ} _{∗}− K_{∗}(1 + T) under the restriction map ℱ_{∗}^{2} R_{ℱ} _{∗}^{2} is zero.

15. Computation of ℛ^{}and ℛ_{}

Let us fix a function χ : → which splits the projection → , namely, take

(15.1)  χ(k mod p) = k mod p^{2}, 0 k < p.

We will use χ to define splittings of ℛ^{}= (ℛ_{1}^{}⟶^{∂} ℛ_{0}^{}) . Here a splitting s of ℛ^{}is an -linear map for which the diagram

(15.2)  [figure: ℛ_{1}^{}; ∂; R_{ℱ}; s; ℱ_{0}; ℛ_{0}^{}]

commutes with R_{ℱ} = im(∂) = ker(q_{ℱ} : ℱ_{0} → 𝒜). We only consider the case p = 2.

(15.3) Definition (The right equivariant splitting of ℛ^{}). Using χ, all Adem relations

[a, b] := Sq^{a}Sq^{b} + \begin{matrix}[ & \frac{a}{2} & ] \\ ∑ \\ k=0 \end{matrix} \binom{b − k − 1}{a − 2k}Sq^{a+b−k}Sq^{k}

for a, b > 0, a < 2b, can be lifted to elements [a, b]_{χ} ∈ R_{ℬ} by applying χ to all coefficients, i. e. by interpreting [a, b] as an element of ℬ. As shown in [1, 16.5.2], R_{ℱ} is a free right ℱ_{0}-module with a basis consisting of preadmissible relations. For p = 2 these are elements of the form

Sq^{a_{1}} ⋯ Sq^{a_{k−1}}[a_{k}, a] ∈ R_{ℱ}

satisfying a_{1} 2a_{2}, ..., a_{k−2} 2a_{k−1}, a_{k−1} 2a_{k}, a_{k} < 2a. Sending such an element to

Sq^{a_{1}} ⋯ Sq^{a_{k−1}}[a_{k}, a]_{χ} ∈ R_{ℬ}

determines then a unique right ℱ_{0}-equivariant splitting ϕ in the pair algebra ℛ^{}; that is, we get a commutative diagram

[figure: R_{ℱ}; ϕ; R_{ℬ} ⊗ = ℛ_{1}^{}; ∂; ℱ_{0}; ℛ_{0}^{}.]

For a splitting s of ℛ^{}the map s⊗1 ⊕ 1⊗s induces the map s_{#} in the diagram

(15.4)  [figure: ℛ_{1}^{}; Δ; ∂; (ℛ^{}ˆ⊗ℛ^{})_{1}; ∂ ˆ⊗; ℛ_{1}^{}⊗ℱ_{0} ⊕ ℱ_{0}⊗ℛ_{1}^{}; R_{ℱ}; s; Δ_{R}; R_{ℱ}^{(2)}; s_{#}; R_{ℱ} ⊗ℱ_{0} ⊕ ℱ_{0}⊗R_{ℱ}; s⊗1⊕1⊗s; ℱ_{0}; Δ; ℱ_{0} ⊗ ℱ_{0}.]

Then the difference U = s_{#}Δ_{R}− Δs : R_{ℱ} → (ℛ^{}ˆ⊗ℛ^{})_{1} satisfies ∂ ˆ⊗U = 0 since

∂ ˆ⊗s_{#}Δ_{R} = Δ_{R} = Δ_{R}∂s = ∂ ˆ⊗Δs.

Thus U lifts to ker∂ ˆ⊗ 𝒜 ⊗ 𝒜 and gives an -linear map

(15.5)  U_{s} : R_{ℱ} → 𝒜 ⊗ 𝒜 .

If we use the splitting s to identify ℛ_{1}^{}with the direct sum 𝒜 ⊕ R_{ℱ} , then it is clear that knowledge of the map U_{s} determines the diagonal ℛ_{1}^{}→ (ℛ^{}ˆ⊗ℛ^{})_{1} completely. Indeed s_{#} yields the identification (ℛ^{}ˆ⊗ℛ^{})_{1} 𝒜 ⊗𝒜 ⊕ R_{ℱ}^{(2)}, and under these identifications Δ : ℛ_{1}^{}→ (ℛ^{}ˆ⊗ℛ^{})_{1} corresponds to a map which by commutativity of (15.4) must have the form

(15.6)  𝒜 ⊕ R_{ℱ} \underrightarrow{( \begin{matrix}Δ_{𝒜} & U_{s} \\ 0 & Δ_{R} \end{matrix} )} 𝒜 ⊗𝒜 ⊕ R_{ℱ}^{(2)}

and is thus determined by U_{s}.

One readily checks that the map U_{s} for s = ϕ in (15.3) coincides with the map U defined in [1, 16.4.3] in terms of the algebra ℬ.

Given the splitting s and the map U_{s}, the only piece of structure remaining to determine the Alg_{𝟙}^{pair}- comonoid structure of ℛ^{}completely is the ℱ_{0}-ℱ_{0}-bimodule structure on ℛ_{1}^{}𝒜 ⊕ R_{ℱ} . Consider for f ∈ ℱ_{0}, r ∈ R_{ℱ} the difference s(f r) − f s(r). It belongs to the kernel of ∂ since

∂s(f r) = f r = f ∂s(r) = ∂(f s(r)).

Thus we obtain the left multiplication map

(15.7)  a_{s} : ℱ_{0} ⊗ R_{ℱ} → 𝒜 .

Similarly we obtain the right multiplication map

b_{s} : R_{ℱ} ⊗ ℱ_{0} → 𝒜

by the difference s(rf) − s(r)f.

(15.8) Lemma. For s = ϕ in (15.3) the right multiplication map b_{ϕ} is trivial, that is ϕ is right equivariant, and the left multiplication map factors through q_{ℱ} ⊗ 1 inducing the map

a_{ϕ} : 𝒜 ⊗ R_{ℱ} → 𝒜 .

Proof. Right equivariance holds by definition. As for the factorization, R_{ℱ} ⊗ R_{ℱ} ℱ_{0} ⊗ R_{ℱ} is in the kernel of a_{ϕ} : ℱ_{0} ⊗ R_{ℱ} → 𝒜, since by right equivariance of s and by the pair algebra property (1.9) for ℛ^{}one has for any r, r′ ∈ R_{ℱ}

s(rr′) = s(r)r′ = s(r)∂s(r′) = (∂s(r))s(r′) = rs(r′).

Hence factoring the above map through (ℱ_{0} ⊗ R_{ℱ} )/(R_{ℱ} ⊗ R_{ℱ} ) 𝒜 ⊗ R_{ℱ} we obtain a map

𝒜 ⊗ R_{ℱ} → 𝒜 .

Summarizing the above, we thus have proved

(15.9) Proposition. Using the splitting s = ϕ of ℛ^{}in (15.3) the comonoid ℛ^{}in the category Alg_{𝟙}^{pair} described in (5.4) is completely determined by the maps

U_{ϕ} : R_{ℱ} → 𝒜 ⊗ 𝒜

and

a_{ϕ} : 𝒜 ⊗ R_{ℱ} → 𝒜

given in (15.5) and (15.7) respectively.

We next introduce another splitting s = ψ for which U_{s} = 0. For this we use the fact that 𝒜_{∗} = Hom(𝒜 , ) and

(15.10)  ℬ_{#} = Hom(ℬ_{0}, )

with ℬ_{0} = T_{}(E_{𝒜} ) both are polynomial algebras in such a way that generators of 𝒜_{∗} are also (part of the) generators of ℬ_{#}.

Using χ in (15.1) we obtain the function

(15.11)  ψ_{χ} : 𝒜_{∗} → ℬ_{#}

(which is not -linear) as follows. Each element x in 𝒜_{∗} is uniquely an -linear combination x = ∑ _{α} n_{α}α where α runs through the monomials in Milnor generators. Such a monomial can be also considered as an element in ℬ_{#} by (10.6) so that we can define

ψ_{χ}(x) = ∑_{α} χ(n_{α})α ∈ ℬ_{#}.

(15.12) Definition (The comultiplicative splitting of ℛ^{}). Consider the following commutative diagram with exact rows and columns

[figure: 𝒜_{∗}; ψ_{χ}; ℱ_{∗}; Hom(R_{ℬ} , ); ℬ_{#}; q; Hom(R_{ℬ} , ); 𝒜_{∗}; _{q_{ℱ} _{∗}}; ℱ_{∗}; j; Hom(R_{ℬ} , ); j_{R}; 𝒜_{∗}; qψ_{χ}; ℛ_{}^{0}; ℛ_{}^{1}]

with the columns induced by the short exact sequence and the rows induced by (7.1). In particular q is induced by the inclusion R_{ℬ} ⊂ ℬ_{0}. Now it is clear that ψ_{χ} yields a map qψ_{χ} which lifts to Hom(R_{ℬ} , ) so that we get the map

qψ_{χ} : 𝒜_{∗} → ℛ_{}^{1}

which splits the projection ℛ_{}^{1} 𝒜_{∗}. Moreover qψ_{χ} is -linear since for all x, y ∈ 𝒜_{∗} the elements ψ_{χ}(x) + ψ_{χ}(y) − ψ_{χ}(x + y) ∈ ℬ_{#} are in the image of the inclusion jq_{ℱ} _{∗} : 𝒜_{∗} ℬ_{#} and thus go to zero under q.

The dual of qψ_{χ} is thus a retraction (qψ_{χ})^{∗} in the short exact sequence

[figure: R_{ℱ}; (qψ_{χ})_{⊥}^{∗}; R_{ℬ} ⊗; π; 𝒜; ι; ℛ_{1}^{}; (qψ_{χ})^{∗}]

which induces the splitting ψ = (qψ_{χ})_{⊥}^{∗} of ℛ^{}determined by

ψ(π(x)) = x − ι((qψ_{χ})^{∗}(x)).

(15.13) Lemma. For the splitting s = ψ of ℛ^{}we have U_{ψ} = 0.

Proof. We must show that the following diagram commutes:

\xymatrix{R_{ℱ} \ar[r]^{Δ_{R}} \ar[d]_{ψ} & R_{ℱ}^{(2)} \ar[d]^{ψ_{#}} \\ R_{ℬ} ⊗ \ar[r]^{Δ} & (ℛ^{}ˆ⊗ℛ^{})_{1}.}

Obviously this is equivalent to commutativity of the dual diagram

[figure: (ℛ_{}ˇ⊗ℛ_{})^{1}; Δ_{∗}; (ψ_{#})_{∗}; Hom(R_{ℬ} , ); ψ_{∗}; _{R_{ℱ}^{(2)}_{∗}}; _{Δ_{R}_{∗}}; _{R_{ℱ} _{∗}}]

which in turn is equivalent to commutativity of

(15.14)  [figure: 𝒜_{∗} ⊗ 𝒜_{∗}; (ψ_{#})_{∗}^{⊥}; δ_{∗}; 𝒜_{∗}; qψ_{χ}; (ℛ_{}ˇ⊗ℛ_{})^{1}; Δ_{∗}; Hom(R_{ℬ} , ).]

On the other hand, the left hand vertical map in the latter diagram can be included into another commutative diagram

[figure: ℱ_{∗}⊗𝒜_{∗} ⊕ 𝒜_{∗}⊗ℱ_{∗}; 1⊗qψ_{χ}⊕qψ_{χ}⊗1; 𝒜_{∗} ⊗ 𝒜_{∗}; ^{\binom{i⊗1}{1⊗i}}; (ψ_{#})_{∗}^{⊥}; ℱ_{∗}⊗ℛ_{}^{1} ⊕ ℛ_{}^{1}⊗ℱ_{∗}; (ℛ_{}⊗ ℛ_{})^{1}]

It follows that on elements, commutativity of (15.14) means that the equality

qψ_{χ}(xy) = i(x)qψ_{χ}(y) + qψ_{χ}(x)i(y)

holds for any x, y ∈ 𝒜_{∗}. By linearity, it is clearly enough to prove this when x and y are monomials in Milnor generators.

For this observe that for any x ∈ 𝒜_{∗} = Hom(𝒜 , ), the element qψ_{χ}(x) ∈ Hom(R_{ℬ} , ) is the unique -linear map making the diagram

[figure: R_{ℬ}; qψ_{χ}(x); ℬ_{0}; ψ_{χ}(x); 𝒜; x; ; ; ]

commute. This uniqueness implies the equality we need in view of the following commutative diagram with exact columns:

[figure: R_{ℬ}; R_{ℬ}^{(2)}; ⊗; ; ; ℬ_{0}; Δ; ℬ_{0} ⊗ ℬ_{0}; ψ_{χ}(x)⊗ψ_{χ}(y); ⊗; ; ; 𝒜; δ; xy; 𝒜 ⊗ 𝒜; x⊗y; ⊗; ; ,]

since when x and y are monomials in Milnor generators, one has ψ_{χ}(xy) = ψ_{χ}(x)ψ_{χ}(y).

Therefore we call ψ the comultiplicative splitting of ℛ^{}. We now want to compute the left and right multiplication maps a_{ψ} and b_{ψ} defined in (15.7). The dual maps (a_{ψ})_{∗} and (b_{ψ})_{∗} can be described by the diagrams

(15.15)  [figure: (R_{ℬ} )_{∗}; m_{∗}^{ℓ}; 𝒜_{∗}; qψ_{χ}; m_{∗}; ℱ_{∗} ⊗ (R_{ℱ} )_{∗}; ℱ_{∗} ⊗ (R_{ℬ} )_{∗}; 𝒜_{∗} ⊗ 𝒜_{∗}; i⊗qψ_{χ}]

and

(15.16)  [figure: (R_{ℬ} )_{∗}; m_{∗}^{r}; 𝒜_{∗}; qψ_{χ}; m_{∗}; (R_{ℱ} )_{∗} ⊗ ℱ_{∗}; (R_{ℬ} )_{∗} ⊗ ℱ_{∗}; 𝒜_{∗} ⊗ 𝒜_{∗}.; qψ_{χ}⊗i]

Here m_{∗} is dual to the multiplication in 𝒜 and m_{∗}^{ℓ} and m_{∗}^{r} are induced by the ℱ_{0}-ℱ_{0}-bimodule structure of R_{ℬ} ⊗ . One readily checks

(a_{ψ})_{∗} = m_{∗}^{ℓ}qψ_{χ} − (i ⊗ qψ_{χ})m_{∗}
(b_{ψ})_{∗} = m_{∗}^{r}qψ_{χ} − (qψ_{χ} ⊗ i)m_{∗}.

We now consider the diagram

[figure: ℬ_{#}; m_{∗}^{}; 𝒜_{∗}; ψ_{χ}; m_{∗}; ℱ_{∗} ⊗ ℱ_{∗}; ℬ_{#} ⊗ ℬ_{#}; 𝒜_{∗} ⊗ 𝒜_{∗}; ψ_{χ}^{⊗}]

Here ψ_{χ}^{⊗} is defined similarly as ψ_{χ} in (15.11) by the formula

ψ_{χ}^{⊗} (∑_{α,β} n_{αβ}α ⊗ β) = ∑_{α,β} χ(n_{αβ})α ⊗ β

where α, β run through the monomials in Milnor generators. Moreover m_{∗}^{}is the dual of the multiplication map m^{}of ℬ_{0} = T_{}(E_{𝒜} ).

(15.17) Lemma. The difference m_{∗}^{}ψ_{χ} − ψ_{χ}^{⊗}m_{∗} lifts to an -linear map ∇_{χ} : 𝒜_{∗} → ℱ_{∗} ⊗ ℱ_{∗} such that one has

(a_{ψ})_{∗} = (1 ⊗ π)∇_{χ}
(b_{ψ})_{∗} = (π ⊗ 1)∇_{χ}.

Here π : ℱ_{∗} R_{ℱ} _{∗} is induced by the inclusion R_{ℱ} ⊂ ℱ_{0}.

Proof. We will only prove the first equality; the proof for the second one is completely similar. The following diagram

[figure: R_{ℬ} _{∗}; j_{R}; m_{∗}^{ℓ}; 𝒜_{∗}; qψ_{χ}; m_{∗}; _{R_{ℬ} _{#}}; m_{#}^{ℓ}; ℬ_{#}; m_{∗}^{}; π; 𝒜_{∗}; ψ_{χ}; m_{∗}; ℬ_{#} ⊗ ℬ_{#}; 1⊗π; 𝒜_{∗} ⊗ 𝒜_{∗}; ψ_{χ}^{⊗}; ℬ_{#} ⊗ R_{ℬ} _{#}; ℱ_{∗} ⊗ ℱ_{∗}; 1⊗π; ℱ_{∗} ⊗ R_{ℬ} _{∗}; j⊗j_{R}; 𝒜_{∗} ⊗ 𝒜_{∗}; i⊗qψ_{χ}; ℱ_{∗} ⊗ R_{ℱ} _{∗}]

commutes except for the innermost square, whose deviation from commutativity is ∇_{χ} and lies in the image of ℱ_{∗} ⊗ ℱ_{∗} ↪ ℬ_{#} ⊗ ℬ_{#}, and the outermost square, whose deviation from commutativity is a_{ψ}_{∗} and lies in the image of ℱ_{∗} ⊗ R_{ℱ} _{∗} ↪ ℱ_{∗} ⊗ R_{ℬ} _{∗}. It follows that (1 ⊗ π)∇_{χ} and a_{ψ}_{∗} have the same image under j ⊗ j_{R}, and since the latter map is injective we are done.

Using the splitting ψ we get the following analogue of (15.9).

(15.18) Proposition. The comonoid ℛ^{}in the category Alg_{𝟙}^{pair} described in (5.4)is completely determined by the multiplication maps a_{ψ} and b_{ψ}. In fact, the identification

ℛ_{1}^{}= 𝒜 ⊕ R_{ℱ}

induced by the splitting s = ψ identifies the diagonal of ℛ^{}with Δ_{𝒜} ⊕ Δ_{R} (see (15.5), (15.6)), and the bimodule structure of ℛ_{1}^{}with

f (α, r) = (f α − a_{ψ}(f, r), f r)
(α, r)f = (αf − b_{ψ}(r, f), rf)

for f ∈ ℱ_{0}, r ∈ R_{ℱ} , α ∈ 𝒜.

16. Computation of the Hopf pair algebra ℬ^{}

The Hopf pair algebra 𝒱 = ℬ^{}in (6.15), given by the algebra of secondary cohomology operations, satisfies the following crucial condition which we deduce from [1, 16.1.5].

(16.1) Theorem. There exists a right ℱ_{0}-equivariant splitting

u : ℛ_{1}^{}= R_{ℬ} ⊗ → ℬ_{1} ⊗ = ℬ_{1}^{}

of the projection ℬ_{1}^{}→ ℛ_{1}^{}, see (6.4), such that the following holds. The diagram

[figure: 𝒜 ⊕_{}Σ𝒜; q; ℬ_{1}^{}; q; ℬ_{0}^{}; 𝒜; 𝒜; ū; ℛ_{1}^{}; u; ℛ_{0}^{}; 𝒜]

commutes, where ūis the inclusion. Moreover in the diagram of diagonals, see (6.5),

[figure: ℬ_{1}^{}; Δ_{ℬ}; (ℬ^{}ˆ⊗ℬ^{})_{1}; Σ𝒜 ⊗ 𝒜; ℛ_{1}^{}; Δ_{R}; u; (ℛ^{}ˆ⊗ℛ^{})_{1}; u ˆ⊗u]

the difference Δ_{ℬ} u − (u ˆ⊗u)Δ_{R} lifts to Σ𝒜 ⊗ 𝒜 and satisfies

ξπ̄ = Δ_{ℬ} u − (u ˆ⊗u)Δ_{R}: \xymatrix{ℛ_{}^{1} \ar@{->>}[r]^{π̄} & R̄ \ar[r]^{ξ} & Σ𝒜 ⊗ 𝒜}

where ξ is dual to ξ_{∗} in (14.8). Here π̄ is the projection ℛ_{}R_{ℱ} R̄. The cocycle ξ is trivial if p is odd.

(16.2) Definition. Using a splitting u of ℬ^{}as in (16.1) we define a multiplication operator

A : 𝒜 ⊗ R_{ℬ} → Σ𝒜

by the equation

A(ᾱ ⊗ x) = u(αx) − αu(x)

for α ∈ ℱ_{0}, x ∈ R_{ℬ} . Thus −A is a multiplication map as studied in [1, 16.3.1]. Fixing a splitting s of ℛ^{} as in (15.2) we define an s-multiplication operator A_{s} to be the composite

A_{s} : \xymatrix{𝒜 ⊗ R_{ℱ} \ar[r]^{1⊗s} & 𝒜 ⊗ R_{ℬ} \ar[r]^{A} & Σ𝒜} .

Such operators have the properties of the following s-multiplication maps.

(16.3) Definition. Let s be a splitting of ℛ^{}as in (15.2) and let U_{s}, a_{s}, b_{s} be defined as in section 15. An s-multiplication map

A_{s} : 𝒜 ⊗ R_{ℱ} → 𝒜

is an -linear map of degree −1 satisfying the following conditions with α, α′, β, β′ ∈ ℱ_{0}, x, y ∈ R_{ℱ} (1) A_{s}(α, xβ) = A_{s}(α, x)β + (α)b_{s}(x, β)

(2) A_{s}(αα′, x) = A_{s}(α, α′x) + (α)a_{s}(α′, x) + (−1)^{deg(α)}αA_{s}(α′, x)
(3) δA_{s}(α, x) = (A_{s})_{#}(α ⊗ Δx) + L(α, x) + ∇_{ξ}(α, x) + δ (α)U_{s}(x).

Here (A_{s})_{#} : 𝒜 ⊗ R_{ℱ}^{(2)} → 𝒜 ⊗ 𝒜 is defined by the equalities

(A_{s})_{#}(α ⊗ x ⊗ β′) = ∑ (−1)^{deg(α_{r}) deg(x)}A_{s}(α_{ℓ}, x) ⊗ α_{r}β′,
(A_{s})_{#}(α ⊗ β ⊗ y) = ∑ (−1)^{deg(α_{r}) deg(β)+deg(α_{ℓ})+deg(β)} α_{ℓ}β ⊗ A_{s}(α_{r}, y),

where as always

δ(α) = ∑ α_{ℓ} ⊗ α_{r} ∈ 𝒜 ⊗ 𝒜 .

Two s-multiplication maps A_{s} and A′_{s} are equivalent if there exists an -linear map

γ : R_{ℱ} → 𝒜

of degree −1 such that the equality

A_{s}(α, x) − A′_{s}(α, x) = γ(αx) − (−1)^{deg(α)}αγ(x)

holds for any α ∈ 𝒜, x ∈ R_{ℱ} and moreover γ is right ℱ_{0}-equivariant and the diagram

\xymatrix{𝒜 \ar[r]^{δ} & 𝒜 ⊗ 𝒜 \\ R_{ℱ} \ar[u]_{γ} \ar[r]_{Δ} & R_{ℱ}^{(2)} \ar[u]_{γ_{#}}}

commutes, with γ_{#} given by

γ_{#}(x ⊗ β) = γ(x) ⊗ β,
γ_{#}(α ⊗ y) = (−1)^{deg(α)}α ⊗ γ(y)

for α, β ∈ ℱ_{0}, x, y ∈ R_{ℱ} .

(16.4) Theorem. There exists an s-multiplication map A_{s} and any two such s-multiplication maps are equivalent. Moreover each s-multiplication map is an s-multiplication operator as in (16.2) and vice versa.

Proof. We apply [1, 16.3.3]. In fact, we obtain by A_{s} the multiplication operator

A : 𝒜 ⊗ R_{ℬ} = 𝒜 ⊗𝒜 ⊕ 𝒜 ⊗R_{ℱ} → Σ𝒜

with A(α ⊗ x) = A_{s}(α ⊗ x̄) + (α)ξ where (x̄, ξ) ∈ R_{ℱ} ⊕ 𝒜 = R_{ℬ} ⊗ corresponds to x, that is s(x̄) + ι(ξ) = x for ι : 𝒜 ⊂ R_{ℬ} ⊗ .

(16.5) Remark. For the splitting s = ϕ of ℛ^{}in (15.3) the maps

A_{n,m}: 𝒜 → 𝒜

are defined by A_{n,m}(α) = A_{ϕ}(α ⊗ [n, m]), with [n, m] the Adem relations in R_{ℱ} . Using formulæ in (16.3) the maps A_{n,m}determine the ϕ-multiplication map A_{ϕ} completely. The maps A_{n,m}coincide with the corresponding maps A_{n,m}in [1, 16.4.4]. In [1, 16.6] an algorithm for determination of A_{n,m}is described, leading to a list of values of A_{n,m}on the elements of the admissible basis of 𝒜. The algorithm for the computation of A_{n,m}can be deduced from theorem (16.4) above.

(16.6) Remark. Triple Massey products ⟨α, β, γ⟩ with α, β, γ ∈ 𝒜, αβ = 0 = βγ, as in (6.16) can be computed by A_{s} as follows. Let β̄γ̄ ∈ R_{ℬ} be given as in (6.16). Then β̄γ̄ ⊗ 1 ∈ R_{ℬ} ⊗ satisfies

β̄γ̄ ⊗ 1 = s(x̄) + ι(ξ)

with x̄∈ R_{ℱ} , ξ ∈ 𝒜 and ⟨α, β, γ⟩satisfies

A_{s}(α ⊗ x̄) + (α)ξ ∈ ⟨α, β, γ⟩.

Compare [1, 16.3.4].

Now it is clear how to introduce via a_{s}, b_{s}, U_{s}, ξ, , and A_{s} a Hopf pair algebra structure on

(16.7)  [figure: 𝒜 ⊕ Σ𝒜 ⊕ R_{ℱ}; q; 𝒜 ⊕ R_{ℱ}; ℬ_{1}^{}; ℛ_{1}^{}]

which is isomorphic to ℬ^{}, compare (15.9).

In the next section we describe an algorithm for the computation of a ψ-multiplication map, where ψ is the comultiplicative splitting of ℛ^{}in (15.12). For this we compute the dual map (A_{ψ})_{∗} of A_{ψ}.

17. Computation of the Hopf pair coalgebra ℬ_{}

For the comultiplicative splitting s = ψ of ℛ^{}in (15.12) we introduce the following ψ-comultiplication maps which are dual to the ψ-multiplication maps in (16.3).

(17.1) Definition. Let a_{ψ}_{∗} and b_{ψ}_{∗} be given as in (15.17). A ψ-comultiplication map

A^{ψ} : 𝒜_{∗} → 𝒜_{∗} ⊗ R_{ℱ} _{∗}

is an -linear map of degree +1 satisfying the following conditions.

(1) The maps in the diagram

[figure: 𝒜_{∗} ⊗ R_{ℱ} _{∗}; 1⊗m_{∗}^{r}; 𝒜_{∗}; m_{∗}; A^{ψ}; 𝒜_{∗} ⊗ R_{ℱ} _{∗} ⊗ ℱ_{∗}; 𝒜_{∗} ⊗ 𝒜_{∗}; A^{ψ}⊗i]

satisfy

(1 ⊗ m_{∗}^{r})A^{ψ} − (A^{ψ} ⊗ i)m_{∗} = ( _{∗} ⊗ b_{ψ}_{∗})m_{∗}.

Here _{∗} is computed in (9.7) and m_{∗}^{r} is defined in (15.16).

(2) The maps in the diagram

[figure: 𝒜_{∗} ⊗ R_{ℱ} _{∗}; m_{∗}⊗1; 𝒜_{∗}; A^{ψ}; A^{ψ}; 𝒜_{∗} ⊗ 𝒜_{∗} ⊗ R_{ℱ} _{∗}; 1⊗i⊗1; 𝒜_{∗} ⊗ ℱ_{∗} ⊗ R_{ℱ} _{∗}; 𝒜_{∗} ⊗ R_{ℱ} _{∗}; 1⊗m_{∗}^{ℓ}]

satisfy

(1 ⊗ i ⊗ 1)(m_{∗} ⊗ 1)A^{ψ} = (1 ⊗ m_{∗}^{ℓ})A^{ψ} + (τ ⊗ i ⊗ 1)(1 ⊗ A^{ψ})m_{∗} + ( _{∗} ⊗ a_{ψ}_{∗})m_{∗}.

Here m_{∗}^{ℓ} is as in (15.15), and τ : 𝒜_{∗} → 𝒜_{∗} is given by τ(α) = (−1)^{deg(α)}α.

(3) For x, y ∈ 𝒜_{∗} the product xy in the algebra 𝒜_{∗} satisfies the formula

A^{ψ}(xy) = A^{ψ}(x)m_{∗}(y) + (−1)^{deg(x)}m_{∗}(x)A^{ψ}(y) + L_{∗}(x, y) + ∇_{ξ} _{∗}(x, y).

Here L_{∗} and ∇_{ξ} _{∗} are given in 12.12 and 14.16 respectively, with L_{∗} = ∇_{ξ} _{∗} = 0 for p odd. Two ψ-comultiplication maps A^{ψ}, Ā^{ψ} are equivalent if there is a derivation

γ_{∗} : 𝒜_{∗} → R_{ℱ} _{∗}

of degree +1 satisfying the equality

A^{ψ} − Ā^{ψ} = m_{∗}^{ℓ}γ_{∗} − (τ ⊗ γ_{∗})m_{∗}.

As a dual statement to (16.4) we get

(17.2) Theorem. There exists a ψ-comultiplication map A^{ψ} and any two such ψ-comultiplication maps are equivalent. Moreover each ψ-comultiplication map A^{ψ} is the dual of a ψ-multiplication map A_{ψ} in (16.4) with A^{ψ} = A_{ψ}_{∗}.

Now dually to (16.7), it is clear how to introduce via a_{ψ}_{∗}, b_{ψ}_{∗}, ξ_{∗}, _{∗}, and A^{ψ} a Hopf pair coalgebra structure on

[figure: 𝒜_{∗} ⊕ Σ𝒜_{∗} ⊕ R_{ℱ} _{∗}; 𝒜_{∗} ⊕ R_{ℱ} _{∗}; i; ℬ_{}^{1}; ℛ_{}^{1}]

which is isomorphic to ℬ_{}, compare (15.18).

18. Algorithms for machine computations

We finally briefly indicate how to use the obtained expressions to perform actual calculations on a computer. Notably, knowledge of either a multiplication map such as A_{ϕ} or of a comultiplication map such as A^{ψ} enables one to calculate all ordinary and matric triple Massey product in the Steenrod algebra, as well as the d_{(2)} differential of the Adams spectral sequence, and hence its E_{3} term — see [1, 2].

Using conditions in (17.1) one obtains an algorithm computing the comultiplication map A^{ψ} as follows. At nth step, n = 1, 2, ..., one picks the Milnor generator ζ_{n} and seeks the value A^{ψ}(ζ_{n}) as a linear combination

∑ ϵ_{a,r}a ⊗ r

where a ⊗ r runs through the basis of the 2^{n}th homogeneous component of 𝒜_{∗} ⊗ R_{ℱ} _{∗} formed by tensors a ⊗ r with deg(a) + deg(r) = 2^{n} where a is a monomial in Milnor generators ζ_{k} and r a monomial in the elementary Lyndon elements M_{d_{1},...,d_{ℓ}}containing at least one element not in {M_{1}, M_{2,1}, M_{4,2,1}, ...}(see description after (11.2)).

The conditions (1) and (2) in (17.1)impose linear equations on the numbers ϵ_{a,r}; solution is not uniquely determined, so one fixes one possible solution and obtains the corresponding value A^{ψ}(ζ_{n}). After that, condition (3) in (17.1) gives all values of A^{ψ} on the subalgebra of 𝒜_{∗} generated by the ζ_{k} for k n. One then moves to step n + 1.

One sees easily that this inductive procedure indeed works, that is, at each step n the conditions (1) and (2) are expressible by values A^{ψ}(ζ_{k}) with k n.

(18.1) Remark. In [1, Chapter 16], an algorithm for computation of the multiplication map A_{ϕ} is presented. This algorithm has been implemented and used in [2] to compute values of the differential d_{(2)} in the Adams spectral sequence up to degree 40. The main computational strain of that algorithm lies in the fact that in the intermediate steps one has to compute auxiliary values of the maps such as ∇_{ϕ} which involve expressions much larger than the final answer.

The dual approach presented here gives an algorithm for A^{ψ} which is more economic in this respect, in the sense that the intermediate values to compute are closer in size to the final value. It is thus expected that with the same computational resources, values in much higher degrees can be reached.

References

[1] Hans-Joachim Baues, The algebra of secondary cohomology operations, 2004.

[2] Hans Joachim Baues and Mamuka Jibladze, The computation of the E_{3}-term of the Adams spectral sequence, arXiv:

    math.AT/0407045.

[3] Hans Joachim Baues, Mamuka Jibladze, and Teimuraz Pirashvili, Strengthening track theories, arXiv: math.CT/0307185.

[4] Hans Joachim Baues and Günther Wirsching, Cohomology of small categories, J. Pure Appl. Algebra 38 (1985), 187–211.

[5] Michiel Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math. 164 (2001), 283–300.

[6] Leif Kristensen, Massey products in Steenrod’s algebra, Proc. Advanced Study Inst. on Algebraic Topology (Aarhus, 1970), Vol. II, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 240–255.

[7] John Milnor, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.

 [8] On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/projects/OEIS?Anum=A059966