Hopf Algebras with Integrals

Phùng Hô` Ha’i

Hanoi Institute of Mathematics, P.O.Box 631, 10000 Boho, Hanoi and

Max-Planck Institut für Mathematik, Gottfried-Claren-Str. 26, 53225 Bonn, Germany

1. Introduction

Integrals on Hopf algebras are motivated by the Haar integrals on compact groups. Hopf algebras possessing an integral are also called co-Frobenius Hopf algebras. Integrals on Hopf algebras were studied by several authors [13, 12, 9, 1, 11]. The main results obtained are the equivalence between the the existence of an integral and the existence of a finite dimensional injective comodule, and the uniqueness of the integral (up to a constant).

In this work, we study the injective hull of simple comodule over a Hopf algebra with integral. Among them there is an interesting class of splitting comodules, that are simple comodules, which coinside with their injective hull. These comodules split in any comodule.

The notion of splitting comodules comes from the representation theory of Lie super algebras, due to V. Kac [7]. Irreducible representations of a simple Lie super algebras split into two classes, the class of typical representations and the class of non-typical representations. One of the characterization of typical representations is that they split in any representation.

Using the integral, we give a criteria for a simple comodule over a Hopf algebra to be a splitting comodule. As an application, we classify simple comodules over the Hopf algebra, associated to a Hecke symmetry of birank (1, 1) [5], i.e., quantum (super) groups of type A_{0,0}.

2. Co-Frobenius Coalgebras and Hopf algebras

We work over a field k. Every tensor product if not explicitly indicated means tensor product over k.

Let C be a coalgebra and M be a right C-comodule, the coaction of C on M is denoted by ρ, ρ : M ⟶ M ⊗ C, ρ(v) = v_{0} ⊗ v_{1}. Let C^{∗} := Hom_{k}(C, k) be the dual of C. Then C^{∗} is a algebra, acting on M in the following way

ϕ ⇀ v := v_{0}ϕ(v_{1}).

Analogously, if λ : N ⟶ C ⊗ N, λ(v) = v_{1} ⊗ v_{2}, is a left C-comodule, then it is a right C^{∗}-module through the action v ↼ ϕ := ϕ(v_{1}) ⊗ v_{2}.

Thus, we have a fuctor from the category of right (left) C-comodules into the category of left (right) C^{∗}-modules, which is full, faithfull and exact. The following statement is due to Doi.

2.1 [1] Let M be a right (left) C-comodule, which is finite dimensional. Then M is injective (projective) if and only if it is injective (projective) as left (right) C^{∗}-module.

A C^{∗}-module may not be a C-comodule by the above correspondence. Those C^{∗}-module induced from C-comodules are called rational modules. Each left C^{∗}- module M contains a unique maximal rational submodule, denoted by_{rat}M. Analogously, for a right C^{∗}-module M, its rational submodule will be denoted by M_{rat}.

Let M be a C-comodule. The map ρ : M ⟶ M ⊗C induce a map M^{∗}⊗M ⟶ C, which can be considered as a coalgebra homomorphism or a morphism of Ccomodules, where C coacts on M^{∗} ⊗ M on the second tensor component. In the latter case, we shall use the notation (M^{∗}) ⊗ M to indicate that C coacts only on M. The image of M^{∗} ⊗ M is called the coefficient space of M, denoted by Cf (M).

Let S be a simple (left or right) C-comodule. The fundamental theorem of comodule (saying that a finite generated comodule is finite dimensional) implies that S is finite dimensional. Let D := End^{C}(S). Then, by Schur lemma, D is a division algebra over k and V is a vector space over D. We have Cf (S) ≅ V^{∗} ⊗_{D} V, as coalgebras [2]. Let {M_{α}|α ∈ A} be the set of all simple C-comodules. We define D_{α} as above and m_{α} := dim_{D_{α}}(M_{α}), d_{α}^{2} := dim_{k}D_{α}. Note that d_{α} are positive integers.

If k is algebraically closed then D_{α} = k, ∀α, i.e., d_{α} = 1.

By definition, the socle of a comodule M is the sum of all its simple subcomodule. The sum is plainly direct, it is denoted by σ(M). The injective hull (or cover) of M, is an injective comodule J (M) to gether with a morphism M ⟶ J (M) inducing an isomorphism σ(M) ⟶ σ(J (M)). It is easy to see that the injective hull of a simple comodule, if it exists, is indecomposable. The following results are due ot J. Green.

2.2 [2] (i) The injective hull of any comodule exists uniquely.

(ii) C itself decomposes in to indecomposable injective subcomodules as follows

C ≅ ⨁_{α∈A} J (M_{α})^{⊕m_{α}}.  (1)

(iii) If C = ⨁_{λ∈L} C_{λ} is another decomposition then for each α ∈ A, the set {λ ∈ L|C_{λ} ≅ J (M_{α})} contains exactly m_{α} elements.

A bilinear form b on C is called balanced if, for all ϕ ∈ C^{∗},

b(x ↼ ϕ, y) = b(x, ϕ ⇀ y).

Balanced bilinear forms on C are in 1-1 corespondence with right C^{∗} comodules homomorphism r : C ⟶ C^{∗} by the formula r(x)(y) = b(x, y) and in 1-1 corespondece with left C^{∗} comodules homomorphism r : C ⟶ C^{∗} by the formula l(x)(y) = b(y, x).

A coalgebra is called left (right) co-Frobenius if there exist a left (right) monomorphism of C^{∗}-modules C ⟶ C^{∗}. The following results are due to B. Lin.

2.3 [9] If C is a left co-Frobenius coalgebra then:

(i) The injective cover of every finite dimensional right C-comodule is finite dimensional.

(ii) Every injective right C-comodule is projective.

(iii) _{rat}C^{∗} is dense in C^{∗}.

In the next section we shall need the following result.

Lemma 2.4 Let M be a C-comodule of finite dimension. Then M is projective (resp. injective) if and only if it is projective (resp. injective) in the category of finite dimensional C-comodules.

Proof. It is sufficient to show the “if” part. Assume that M is projective in the category of finite dimensional C-comodule. Consider a diagram

M
∃ ↙ ↓ π
N ⟶^{ν} P ⟶ 0.

By replacing P with Im(P) and N with ν^{−1}(Im(P)), we can assume that π is surjective, thus P is finite dimensional. Let P be a basis of P and N be a set of elements of N such that ν(N) = P. the submodule N_{1} of N, generated by N is finite dimensional and we have ν(N_{1}) = P. Hence, by assumption, there exists a morphism μ : M ⟶ N_{1}: ν ∘ μ = π.

Assume that M is injective in the category of finite dimensional C-comodule. Consider the diagram

0 ⟶ P ⟶^{ν} N
π ↓ ↙ ∃
M.

By replacing P, N with P/Ker(π), N/Ker(π), we can assume that π injective. Hence P has finite dimension. If N/P is finite dimensional, we are done by the assumption on M. Otherwise, consider the set

A := {(N_{α}, μ_{α})|N_{α} ⊃ P, μ_{α} : N_{λ} ⟶ M, μ_{λ} ∘ ν = π} .

Define an order on this set, setting α ≺ β iff N_{α} ⊂ N_{β} and μ_{β}|_{N_{β}} = μ_{α}. The chain condition is satisfied hence there exists a maximal element, say, N_{1}. Since any submodule of N, containing P and having finited dimension is contained in A, N_{1} is strictly bigger then P. Were N_{1} ≠ N, repeating the above process, we would get a submodule N_{1}, N_{2} ≻ N_{1}, which contradicts the maximality of N_{1}. Hence N_{1} = N.

Let H be a Hopf algebra. Then k is a left (right) H-comodule be means of the unit map. A left (right) integral on H is an H-comodule morphism H ⟶ k, where H is considered as left (right) comodule on itself by means of the copoduct.

Let ∫_{r} (resp. ∫_{l}) denoted a left (resp. right) integral on H, then we have

a_{1} ∫_{l} (a_{2}) = ∫_{l} (a) ,  (2)
∫_{r} (a_{1}) a_{2} = ∫_{r} (a) ,  (3)

∀a ∈ H.

We need the following information on the integrals.

2.5 Let H be a Hopf algebra. The following conditions are equivalent (i) H possesses a left integral. (ii) H is left co-Frobenius as a coalgebra. (iii) H possesses a right integral. (iv) H is right co-Frobenius as a coalgebra.

(v) The injective hull of every left comodule is finite dimension. (vi) The injective hull of every right comodule is finite dimension. (vii) H possesses a finite dimension injective left comodule. (viii) H possesses a finite dimension injective right comodule. (ix) H_{rat}^{∗} is dense in H^{∗}.

(x) _{rat}H^{∗} is dense in H^{∗}. (xi) Every injective left H-comodule is projective. (xii) Every injective right H-comodule is projective.

The first 6 conditions are due to Larson-Sweedler-Sullivan, the conditions (vii)-(xii) follow from 2.3

Define bilinear form b: b(x, y) := ∫_{l} (xS(y)) . Using the identity

h_{1} ∫_{l} (h_{2}S(g)) = ∫_{l} (hS(g_{1})) g_{2}  (4)

which follows immediately form the definition of ∫_{l}, we can easily show that b is balanced. The folowing results are due to Ḑ. Ştefan.

2.6 [11] Let H be a Hopf algebra with integral. Then the following facts hold. (i) The bilinear form b is non-degenerate. (ii) For any finite dimensional H-comodule, dim_{k}(Hom^{H}(H, M)) = dim_{k}M.

In particular, we have (i) the antipode is injective, and (ii) there exists h such that ∫_{l} (h) ≠ 0. Since ∫_{l} ∘S satisfies (3), it is a right

integral on H.

Assume for a moment that the field k is algebraically closed. Let R be the radical of H, i.e. R = ⊕Cf (M_{α}), where {M_{α}, α ∈ A} is the set of all simple left (or right) H-comodules. As we have seen in the previous subsection, Cf (M_{α}) ≅ M(m_{α})^{∗}, m_{α} := dim_{k}(M_{α}). Fix idempotents {e_{α,i}|α ∈ A, 1 ≤ i ≤ m_{α}} of the algebras M(m_{α}). They can be considered as linear functional on R by defining

e_{α,i}(Cf (M_{β})) = 0, whenever α ≠ β.

A theorem of Sweeder-Sullivan [12], stating that there exists a coalgebra projection H ⟶ R, implies e_{α,i} can be extended on the whole H and that

H ↼ e_{α,i} ( resp. e_{α,i} ⇀ H) is a right (resp. left) H-comodules.

Consequently, we have a decompostion

H ≅ ⨁_{{α∈A, \atop 1≤i≤mα}} H ↼ e_{α,i} ( resp. ≅ ⨁_{{α∈A, \atop 1≤i≤mα}} e_{α,i} ⇀ H)

as right (resp. left) H-comodules.

On the other hand, it is easy to see that M_{α} ⊂ H ↼ e_{α,i} as right H-comodules. Thus, comparing with the decomposition in 2.2, we have:

2.7 Assume that the field k is algebraically closed. Then

J (M_{α}) ≅ H ↼ e_{α,i}  (5)

as right H-comodules.

3. The Injective Hull of a Simple Comodule

We define a new product on H:

g ∗ h := h_{1} ∫_{l} (h_{2}S(g)) = ∫_{l} (hS(g_{1})) g_{2} (by (4)).

Using 4 we can easily check that ∗ is associative. Denote Ȟ := (H, ∗). Then Ȟ is a (non-unital) algebra. Let V be a right H-comodule. Then V is a left H-module by means of the action

h ∗ v := v_{0} ∫_{l} (v_{1}S(h)) .

The verification again uses (4). Denote V̌ := (V, ∗).

Let f : V ⟶ W be a homomorphism of right H-comodules, i.e., f(v)_{0} ⊗ f(v)_{1} = f(v_{0}) ⊗ v_{1}. We have

h ∗ f(v) = f(v)_{0} ∫_{l} (f(v)_{1}S(h)) = f(v_{0}) ∫_{l} (v_{1}S(h)) = f(h ∗ v).

Thus f is a homomorphism of left Ȟ-modules. Conversely, if f is a homomorphism V̌ ⟶ W̌, then we have, for all h ∈ H,

f(v)_{0} ∫_{l} (f(v)_{1}S(h)) = f(v_{0}) ∫_{l} (v_{1}S(h)) .

By the non-degeneracy of the integral (2.6, (i)), we have

f(v)_{0} ⊗ f(v)_{1} = f(v_{0}) ⊗ v_{1},

which means that f is a homomorphism of right H-comodules. Thus we have

HomȞ(V̌ , W̌) = Hom^{H}(V, W).  (6)

In particular we have proved

Lemma 3.1 If M is a simple right H-comodule then M̌ is a simple left Ȟ-module.

Let now V be a cyclic Ȟ-module, that is, there exists v̄ ∈ V, such that V is generated by v̄. We want to define a coaction of H on V. Let v ∈ V. Then there exists (not uniquely) h ∈ Ȟ, such that v = h ∗ v̄. Set δ̂(v) := h_{1} ∗ v̄ ⊗ h_{2}. We show that δ̂ is independent of the choice of v̄ and h, and that it is in fact a coaction of H on V.

The fact that δ̂ does not change when v̄ is replaced by ṽ is represented by the equation

h_{1} ∗ v̄ ⊗ h_{2} = (h ∗ g)_{1} ∗ ṽ ⊗ (f ∗ g)_{2},

where g ∗ ṽ = v̄. This equation follows immediately form the definition of ∗. The independence on the choice of h means that, whenever h ∗ v̄ = 0, we have h_{1} ∗ v̄ ⊗ h_{2} = 0. Indeed, we have

∫_{l} (h_{2}S(g)) h_{1} ∗ v̄ = (g ∗ h) ∗ v̄ = g ∗ (h ∗ v̄) = 0,

for all g ∈ H. By the non-degeneracy of ∫ , we conclude that h_{1} ∗ v̄ ⊗ h_{2} = 0.

The coassociativity and counitary of δ̂ are also checked directly using (4). Moreover, denoting by V̂ the resulting H-comodule, we also have ˇ V̂ ≅ V.

Since simple modules are cyclic, we have

Lemma 3.2 Let M be a simple Ȟ-module, then there exists a right H-comodule M̂, such that M ≅ ˇ M̂.

We also need another action of Ȟ on a right comodule V of H, given by

h ∘ v := v_{0} ∫_{l} (hS(v_{1})) .

Indeed, we have

g ∘ (h ∘ v) = g ∘ v_{0} ∫_{l} (hS(v_{1})) = v_{0} ∫_{l} (gS(v_{1})) ∫_{l} (hS(v_{2}))
= v_{0} ∫_{l} (hS (∫_{l} (gS(v_{1})) v_{2})) = v_{0} ∫_{l} (hS(g_{1})) ∫_{l} (g_{2}S(v_{1}))
= (g ∗ h) ∘ v.

It is again easy to checke that V̆ := (V, ∘) is a left Ȟ-module, and that, if V is simple then V̆ is simple. Lemma 3.2 holds only in case the antipode is bijective. In fact, if V is a cyclic Ȟ-module. Then we can define the following coaction of H on V:

δ̇(v) := h_{1} ∘ v̄ ⊗ S^{−2}(h_{2}),

where v̄ is a generating element and h ∗ v̄ = v. Denote by V̇ the comodule of H induced from V, we have ˙ V̆ ≅ V.

Composing the operation ˘ and ˆ on a simple comodule V we obtain a new simple comodule ˆ V̆, denoted by V^{•}. The coaction of H on V^{•} is given by

δ^{•}(v) = v_{0} ∫_{l} (h_{1}S(v_{1})) ⊗ h_{2},

with h given by condition v_{0} ∫_{l} (hS(v_{1})) = v.

Now, assume that k is algebraically closed. Let M_{α} be a simple right Hcomodule. Then M̆_{α} is a simple left Ȟ-comodule. The action of Ȟ on M̆_{α} induces a Ȟ-module homomorphism π : Ȟ ⟶ M̆_{α} ⊗(M̆_{α}^{∗}), where M̆_{α} ⊗(M̆_{α}^{∗}) ≅ M̆_{α}^{⊕dim_{k}M_{α}} as Ȟ-modules. The isomorphism (6) shows that π is a homomorphism of Hcomodules.

On the other hand, H decomposes into the direct sum of its indecomposable injective subcomodules as in Lemma 2.2. For h ∈ H ↼ e_{β,j}, i.e., h = e_{β,j}(g_{1})g_{2} for some g ∈ H, and for v ∈ M̆_{α}, we have

h ∗ v = v_{0}e_{β,j}(g_{1}) ∫_{l} (g_{2}S(v_{1})) = v_{0} ∫_{l} (g(S(v_{1})) e_{β,i}(v_{2}).

Thus, if α ≠ β, h ∗ M_{α} = 0, therefore, π(h) = 0. Thus

Hom^{H}(J (M_{α}), M_{α}^{•}) ≠ 0.  (7)

According to 2.6,

dim_{k}(Hom^{H}(H, M_{α}^{•})) = dim_{k}M_{α}^{•}.  (8)

Since H contains precisely m_{α} = dim_{k}M_{α} copies of J (M_{α}), we conclude that

dim_{k}Hom^{H}(J (M_{α}), M_{α}^{•}) = 1,  (9)
dim_{k}Hom^{H}(J (M_{β}), M_{α}^{•}) = 0 if α ≠ β.  (10)

Theorem 3.3 Let H be a Hopf algebra with integral. Then for any simple comodules M_{α}, M_{β}

dim_{k}Hom(J (M_{α}), M_{β}^{•}) = δ_{β}^{α}d_{β}^{2},

where d_{β}^{2} is the dimension over k of D_{β} = End^{H}(M_{β}).

Proof. The case k is algebraically closed is already proved. Assume that k ≠ \overline{k}. Then D_{β} = End^{H}(M_{β}) splits over \overline{k}: D_{β} ⊗_{k} \overline{k} ≅ M\overline{k}(d_{β}). For the extension \overline{H} := H ⊗_{k} \overline{k}, the comodule \overline{M_{β}} := M_{β} ⊗_{k} \overline{k} decomposes into d_{β} exemplars of the simple \overline{H}-comodule M′_{β}. Since \overline{J (M_{β})} := J (M_{β}) ⊗_{k} \overline{k} remains a direct summand of \overline{H}, it is an injective \overline{H}-comodule. Therefore \overline{J (M_{β})} is a direct sum of d_{β} exemplars of J (M′_{β}). Since, for α ≠ β,

Hom\overline{H}(J (M′_{α}), M_{β}^{•}′) = 0,

we have

Hom_{H}(J (M_{α}), M_{β}^{•}) = 0.

Therefore, by virtue of Equation (8) (which is valid on any field),

dim_{k}Hom_{H}(J (M_{α})^{⊕m_{α}}, M_{α}) = dim_{k}M_{α} = d_{α}^{2}m_{α}.

Consequently

dim_{k}Hom_{H}(J (M_{α}), M_{α}) = d_{α}^{2}.

Let M be a finite dimensional right H-comodule then M^{∗} := Hom_{k}(M, k) is also a right comodule with the coaction give by the equation

ρ(ϕ)(x) := ϕ_{0}(x)ϕ_{1} = ϕ(x_{0})S(x_{1}), x ∈ M, ϕ ∈ M^{∗}.

The map ev : M^{∗} ⊗ M ⟶ k, ϕ ⊗ x ⟼ ϕ(x) is a morphism of H-comodules. The pair (M^{∗}, ev) is called left dual to M, it is defined uniquely upto isomorphism. There exists a monomorphism db : k ⟶ S ⊗ S^{∗}, defined by the conditions (ev ⊗ id_{S^{∗}})(id_{S^{∗}} ⊗ db) = id_{S^{∗}} and (id_{s} ⊗ ev)(db ⊗ id_{S}) = id_{S}, which is also a comodule morphism. Dually, (M, ev) is called the right dual to M^{∗}.

Thus, we see that the left dual to a finite dimensional comodule always exists. If the antipoded is bijective then the right dual to any finite dimensional comodule also exists. We shall need the following isomorphism, given by manipulating the morphism ev and db: for any finite dimensional comodule N,

Hom^{H}(M ⊗ N, P) ≅ Hom^{H}(M, P ⊗ N^{∗})  (11)
Hom^{H}(M, N ⊗ P) ≅ Hom^{H}(N^{∗} ⊗ M, P).  (12)

As an immediate corollary of Lemma 2.4 and these equations, we have

Lemma 3.4 Let M be a finite dimensional comodule. Then we have: (i) If M^{∗} is projective (resp. injective) then M is injective (resp. projective). (ii) If the antipode is bijective, then M is injective (resp. projective) iff M^{∗} is

projective (resp. injective).

Proof. Equations (11) and (12) imply

Hom^{H}(M^{∗}, N^{∗}) ≅ Hom^{H}(N, M).  (13)

Thus, if M is projective (resp. injective) then M^{∗} is injective (resp. projective) in the category of finite dimensional comodules.

Corollary 3.5 Let H be a Hopf algebra wiht integral. Assume that the antipode is injective. Then for any simple comodule M

J ((M^{•})^{∗}) ≅ J (M)^{∗}.

Lemma 3.6 Assume that the Hopf algebra H as a left-right integral. Then V^{•} ≅ V^{∗∗}. If H is moreover coquasitriangular then V^{•} ≅ V.

Proof. Assume that ∫ is a left-right integral. Thus we can define V^{•} as above. We want to show that

v_{0} ∫_{l} (h_{1}S(v_{1})) ⊗ h_{2} = c · v_{0} ⊗ S^{2}(v_{1}),

for certain constant c, depending only on ∫ . By the non-degeneracy of integral, this equation is equivalent to

v_{0} ∫ (v_{1}S(h)) ∫ (h_{2}S(g)) = c · v_{0} ∫ (S^{2}(v_{1})S(g)).

We have

v_{0} ∫ (h_{1}S(v_{1})) ∫ (h_{2}S(g))

= v_{0} ∫ ((g ∗ h)S(v_{1})) = (g ∗ h) ∘ v = g ∗ (h ∘ v) = g ∘ v = v_{0} ∫ (gS(v_{1})).

By the uniqueness of integral, we can choose c such that c · ∫ (S^{2}(v_{1})S(g)) =

∫ (gS(v_{1})).

If H is coquasitriangular then V ≅ V^{∗∗}.

Corollary 3.7 With the assumption of Lemma 3.6, we have

J (M^{∗}) ≅ J (M)^{∗}.

Remark 3.8 If the left and the right integrals do not coincide then in general, M^{•} ≅ M̸^{∗∗}. An example is Sweedler’s Hopf algebras, see, e.g., [8].

4. Splitting Comodules

Let S be a simple comodule over H. S is called splitting comodule, or typical comodule, if S = J (S). Since J (S) is injective and hence projective, we see that S splits in any comodule. This explain the name splitting. The name typical was used by V. Kac for modules over a super Lie algebra [7].

By virtue of conditions in 2.5, if a Hopf algebra possesses a splitting comodule then it possesses a non-zero integral. The converse statement is not true. The aim of this section is to give a criteria for a simple comodule to be typical.

Let M is a right H-comodule then the coaction of H on M^{∗∗} – the double left dual to M is given by (identifying M^{∗∗} with M as vector spaces)

ρ_{M^{∗∗}}(v) = x_{0} ⊗ S^{2}(v_{1}).

Lemma 4.1 Let M be a simple H-comodule. Then M is splitting iff M^{∗} is splitting.

Proof. Assume that M is a typical. Then M is injective. By a theorem of Doi [1], M^{∗} ⊗ M is injective too. By definition of M^{∗}, we have an epimorphism ev : M^{∗} ⊗ M ⟶ k. Among indecomposable injective subcomodule of M^{∗} ⊗ M there exists one, say J, such that the restriction of ev on J is not zero. On the other hand, since J is indecomposable and injective, it should appear in the decomposition (1), and by Theorem 3.3, the only comodule with this property is J (k), the injective hull of k. Thus we show that M^{∗} ⊗ M contains J (k) as a subcomodule, consequently, it contains k as subcomodule, i.e. Hom^{H}(k, M^{∗} ⊗ M) ≠ 0. According to (12), we have

Hom^{H}(M^{∗∗}, M) ≠ 0.

Therefore M^{∗∗} ≅ M and hence is splitting. Consequently M^{∗} is also splitting, by Lemma 3.4.

Assume now that M^{∗} is splitting. The discussion above shows that M^{∗∗} is also splitting and M^{∗∗} ≅ M^{∗∗∗∗}. Since

Hom^{H}(M^{∗}, N^{∗}) ≅ Hom^{H}(N, M),

we conclude that S ≅ S^{∗∗}. Thus S is splitting.

Theorem 4.2 Let M be a simple right H-comodule. Then M is splitting if and only if the bilinear form c, c(x, y) = ∫_{r} (yS(x)), is not identically zero on Cf (M).

In this case, c is also non-degenerate on Cf (M).

Proof. “if”.

For each g ∈ H, define a linear functional ϕ_{g} ∈ H^{∗}: ϕ_{g}(h) := ∫_{r} (hS(g)). By

assumption, there exist g ∈ Cf (M) such that ϕ_{g} is not identically zero onCf (M). Since M is simple, the right coideal generated by g is isomorphic to M. Define a linear map η = η_{g}:

η : Cf (M) ⟶ M^{∗∗}, h ⟼ g_{1} ∫_{r} (hS(g_{2})) , h ∈ M.  (14)

Since ε(η(h)) = ϕ_{q}(h), η is not trivial.

We have the following identity, which is an immediately consequence of (3) and the injectivity of the antipode

∫_{r} (h_{1}S(g)) h_{2} = S^{2}(g_{1}) ∫_{r} (hS(g_{2})) .  (15)

It follows form (15) that η is a morphism of H-comodules. Since η is nontrivial on Cf (M), which is a direct sum of copies of M, we conclude that η should induce a morphism M ⟶ M^{∗∗}, which is non-trival. Since M is simple and dim_{k}M = dim_{k}M^{∗∗}, this morphism is an isomorphism. As a consequence, Cf (M^{∗∗}) = S(Cf (M)) = Cf (M) and ϕ_{g} is ∗-invertible on Cf (M). Let ψ be the ∗-inverse to ϕ, define on Cf (M), thus ϕ(h_{1})ψ(h_{2}) = ψ(h_{1})ϕ(h_{2}) = ε(h).

Let now M ↪ N be an inclusion of H-comodules. Let f : N ⟶ M be a linear projection on M. We define a new map F : N ⟶ M as follows,

F (v) := f(v_{0})_{0}ψ(f(v_{0})_{1}) ∫_{r} (v_{1}S(f(v_{0})_{2})) .

F is well defined by the assumption that Im(f) = M, which implies f(v_{0})_{1} ∈ Cf (M). For v ∈ M, f(v) = v, hence

F (v) = v_{0}ψ(v_{1}) ∫_{r} (v_{3}S(v_{2})) = v_{0}ψ(v_{1})q(v_{2}) = v.

Thus, F is again a projection of M. We want to show that F is a morphim of H-comodule, then we will be done.

By defintion of ϕ, we can consider F as a composition of the map g : N ⟶ M^{∗∗}:

g(v) = f(v_{0})_{0} ∫_{r} (v_{1}S(f(v_{0})_{1}) ,

and the morphism η^{−1} : M^{∗∗} ⟶ M. Thus, it is sufficient to show that g is a morphism of H-comodules, which means

f(v_{0})_{0} ∫_{r} (v_{1}S(f(v_{0})_{1}) ⊗ v_{2} = f(v_{0})_{0} ∫_{r} (v_{1}S(f(v_{0})_{2}) ⊗ S^{2}(f(v_{0})_{1}).

We have, according to (15),

the left-hand side = f(v_{0})_{0} ⊗ S^{2}(f(v_{0})_{1}) ∫_{r} (v_{1}S(f(v_{0})_{2}))
= the right-hand side .

Therefore, F is a morphism of H-comodules, consequently, M is injective and J (M) = M.

“only if” Assume now that M is splitting, then, by Lemma 4.1, M^{∗} is also splitting, hence M ⊗M^{∗} is injective. By definition of M^{∗}, there exists a monomorphism db : k ⟶ M ⊗ M^{∗}, inducing a monomorphism J (k) ↪ M ⊗ M^{∗}. The latter inclusion induces the following inclusion

J(k) ⊂ Cf (J (k)) ↪ Cf (M ⊗ M^{∗}) = Cf (M) · Cf (M^{∗}).

Since the right integral does not vanish identically on J (k) (by 3.3), we conlude

that the set ∫_{r} (a_{j}^{i}S(a_{l}^{k})) is not identically zero, as a_{j}^{i}S(a_{l}^{k}) span Cf (M ⊗ M^{∗}).

5. Simple Representations of The Quantum Groups GL_{q}(1|1)

Let V be a finite dimensional vector space over k, a field of characteristic zero, and R be a Hecke symmetry of birank (1, 1). We are interested in simple comodules of the Hopf algebra H, associated to R. The reader is refered to [5] for definitions and known results.

Let E be the bialgebra associated to R. Then simple E-comodules are known from the birank of R. In particular, if birank of R is (1, 1), simple E-comodules can be labeled by pairs (m, n), m ≥ 1, n ≥ 0 and (0, 0). On the other hand, simple E-comodules are also simple as H-comodules with the natural action induced from the inclusion E ⟶ H [4].

Let I_{m,n} be the simple E-comodules associated to the pair (m, n). They are given by the following rule. I_{n,0} = S_{n} is the n-th quantum symmetric tensor over V, I_{1,n−1} = Λ_{n} is the n-th quantum anti-symmetric tensor over V, I_{0,0} := k,

I_{p,q} ⊗ I_{m,n} = I_{m+p,n+q} ⊕ I_{m+p−1,n+q+1},  (16)

for m, p ≥ 1, n, q ≥ 0.

The Koszul complex plays the central role in studying simple E-comodules. It is shown that, if k − l ≠ −rank_{q}R, then the sequence

⋯ ⟶ Λ_{k} ⊗ S_{l}^{∗} ⟶^{d_{k,l}} Λ_{k+1} ⊗ S_{l+1}^{∗} ⟶^{d_{k+1,l+1}} Λ_{k+2} ⊗ S_{l+2}^{∗} ⟶ ⋯

with the differential induced from the dual basis map ev : k ⟶ V ⊗ V^{∗}, is exact [3, 10].

We now show that rank_{q}R = 0. First, notice that, according to (16),

Hom^{H}(I_{p,q}, I_{m+p,n+q} ⊗ I_{m,n}^{∗}) = k
End^{H}(I_{m+p,n+q} ⊗ I_{m,n}^{∗}) = End^{H}(I_{m+p,n+q} ⊗ I_{m,n}) = k^{⊕2}.

Therefore, denoting I_{−m,−n} := I_{m,n}^{∗}, we have

I_{p,q} ⊕ I_{p+1,q−1} ⊂ I_{m+p,n+q} ⊗ I_{−m,−n},  (17)

and I_{m+p,n+q} ⊗ I_{−m,−n} does not contain any other simple E-comodule. Assume that rank_{q}R ≠ 0. Then the complex

0 → k → I_{1,0} ⊗ I_{−1,0} → I_{1,1} ⊗ I_{−2,0} → ⋯  (18)

is exact. We have

I_{n,1} ⊗ I_{1,m−1} ⊗ I_{−m,0} = (I_{n+1,m} ⊕ I_{n,m+1}) ⊗ I_{−m,0}
⊃ I_{n−m+1,m}^{⊕2} ⊕ I_{n−m,m+1} ⊕ I_{n−m+2,m−1}. (19)

Thus, multiplying (18) with I_{n,1}, we have two sequences

\begin{aligned}0 → I_{n,1} → & I_{n,1}^{⊕2} ⊕ I_{n−1,2} ⊕ I_{n+1,0} & & I_{n−1,2}^{⊕2} ⊕ I_{n−2,3} ⊕ I_{n,1} & … \\ & ⋂ & & ⋂ \\ 0 → I_{n,1} → & I_{n,1} ⊗ I_{1,0} ⊗ I_{−1,0} & → & I_{n,1} ⊗ I_{1,1} ⊗ I_{−2,0} & → ⋯ \end{aligned} (20)

The exactness of the lower complex and the remark following (17) imply that I_{n+1,0} = 0, contradiction. Thus we have rank_{q}R = 0. As a consequence, the Hopf algebra H possesses an integral [5] and the formula for the integral in [6] implies that I_{m,n} are all splitting, except for I_{0,0} = k. Therefore, the inclusion in (17) is in fact an isomorphism, by means of the two isomorphism preceeding (17)

I_{p,q} ⊕ I_{p+1,q−1} = I_{m+p,n+q} ⊗ I_{−m,−n}.  (21)

On the other hand, rank_{q}R = 0 implies that all the sequences

⋯ ⟶ I_{1,k−1} ⊗ I_{−l,0} ⟶^{d_{k,l}} I_{1,k} ⊗ I_{−l−1,0} ⟶^{d_{k+1,l+1}} ⊗I_{−l−2,0} ⟶ ⋯  (22)
⋯ ⟶ I_{1,k} ⊗ I_{−k−1,0} ⟶^{d_{l+1,k+1}} ⊗I_{−k−2,0} ⟶^{d_{k+2,l+2}} ⊗I_{−k−3,0} ⟶ ⋯,  (23)

are exact for k > l ≥ 0.

Setting I_{1−l,k−1} := Imd_{k,l} = Kerd_{k+1,l+1}, we have a short exact sequence

0 ⟶ I_{1−l,k−1} ⟶ I_{1,k} ⊗ I_{−l−1,0} ⟶ I_{−l,k} ⟶ 0  (24)

For l = 0, the second and third term in (24) are injective, hence, so is the fourth term. Using induction we can easily show that I_{−l,k} are injective for all k > l ≥ 0. Similar discussion shows that I_{−k,l} := Imd_{l,k} are also injective for all k > l ≥ 0. Thus, we have

I_{1,k} ⊗ I_{−l−1,0} = I_{1−l,k−1} ⊕ I_{−l,k}  (25)
I_{1,l} ⊗ I_{−k−1,0} = I_{−k,l} ⊕ I_{−k−1,l+1}.  (26)

Note that these modules may not be simple.

Denote, for k > l ≥ 0, I_{k,−l} := I_{k,−l}^{∗}, I_{l,−k} := I_{l,−k}^{∗}.

The next step is to define the comodules I_{−1,1} and I_{1,−1}.

Since rank_{q}R = 0, I_{1,0} ⊗ I_{−1,0} = V ⊗ V^{∗} contains two examplars of k in its composition series. Let M := (V ⊗ V^{∗})/k, then the map V ⊗ V^{∗} ⟶ k factorizes through k to a map M ⟶ k. Dualizing this we get a sequence k ⟶ M^{∗} ⟶ V ⊗ V^{∗}. Since Hom^{H}(k, I_{1,1} ⊗ I_{−2,0}) ≅ Hom^{H}(I_{2,0}, I_{1,1}) = 0, Imd_{1,1} ≠ k. From (20), we see that Kerd_{2,2} {⊃ \atop ≠}k. Let N := V ⊗ V^{∗}/Kerd_{2,2}. Then N is a factor comodule of M, which is different from k and M. Therefore V ⊗ V^{∗} contains at least 4 simple comodules in its composition series. It cannot be larger then 4, as on the left hand side of (19), there are 4 simple comodules.

Tensoring (18) with I_{1,0}, we see that there exists a simple comodule in V ⊗V^{∗}, denoted by I_{−1,1}, such that I_{−1,1} ⊗ I_{1,0}. Using induction we can show that

I_{−1,1} ⊗ I_{m,n} = I_{m−1,n+1}, m ≥ 1, n ≥ 0.  (27)

Since V ⊗ V^{∗} is selfdual, the dual to I_{−1,1} is also contained in V ⊗ V^{∗}. We have I_{−1,1} ⊗ I_{1,−1} = k, and

I_{1,−1} ⊗ I_{m,n} = I_{m+1,n−1}, m ≥ 1, n ≥ 0.  (28)

Denote I_{−m,m} := I_{−1,1}^{⊗m}, m ∈ ℤ. Thus, we have associated to each pair (m, n) of integers a comodule I_{m,n} of H. Note that I_{−1,1} is not splitting, whence so are all its powers. Therefore V ⊗ V^{∗} is the injective hull of k. Tedious computations show that

I_{−1,1} ⊗ I_{m,n} = I_{m−1,n+1}, m, n ∈ ℤ.  (29)

From this isomorphism, it follows that the comodules I_{m,n} are all simple. There are no more simple comodules, for each simple comodule should appear in the decomposition series of V^{⊗n} ⊗ V^{∗⊗m}, some m, n ∈ ℕ ∩ 0.

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