ON THE COHOMOLOGY RING OF AN ALGEBRA

Ryszard Nest University of Copenhagen and Boris Tsygan* Pennsylvania State University March 1, 1996

To Yu. I. Manin on his 60th birthday

\underline{Abstract}

We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum multiplication on Floer cohomology of free loop spaces. We discuss some examples, as well as applications to index theorems, characteristic classes and deformations.

0. \underline{Introduction}.

This paper is a continuation and extension of [NT2].

Let X be a smooth manifold or an affine algebraic variety. Let A be the algebra of smooth or regular functions on X. It is well known that most geometric properties of X can be recovered from the algebra A. By (affine) noncommutative geometry one usually means extending corresponding constructions to the case when A is not necessarily commutative.

Among differential-geometric objects, the easiest ones to generalize to the noncommutative setting are vector fields. A noncommutative vector field is just a

Typeset by AMS-TEX

*Partially  supported  by  NSF  Grant  DMS–9307927  .

derivation of an algebra A. As for noncommutative multivector fields and noncommutative differential forms, the answer has been suggested in [HRK]. Consider the standard cochain complex C^{∗}(A, A) computing the Hochschild cohomology

H^{∗}(A, A) = Ext_{A⊗A^{o}}^{∗}(A, A)

(Section 1) and the standard chain complex computing the Hochschild homology

HH_{∗}(A) = H_{∗}(A, A) = T or_{∗}^{A⊗A^{o}}(A, A)

(Section 2). It was shown in [HRK] that, if A is the algebra of regular functions on an affine algebraic variety X over a field k of characteristic zero, then

HH_{i}(A) ≃ Ω_{X/k}^{i}  (0.1)
H^{i}(A, A) ≃ Γ(X, ∧^{i}T X)  (0.2)

The standard cochain complex C^{∗}(A, A) is a differential graded associative algebra (with the cup product which can also be interpreted as the Yoneda product). It is also a differential graded Lie algebra (with the Gerstenhaber bracket [G]). Under the isomorphism (0.1) these operations turn into the wedge product and the Schouten bracket on multivector fields. The cap product from [CE]

C^{p}(A, A) ⊗ C_{n}(A, A) → C_{n−p}(A, A)  (0.3)

becomes a noncommutative analogue of contracting a form with a multivector field.

On the other hand, the standard chain complex C_{∗}(A, A) is not a differential graded algebra unless A is commutative. Instead, there is the exterior product [CE]

C_{∗}(A, A) ⊗ C_{∗}(B, B) → C_{∗}(A ⊗ B, A ⊗ B)  (0.4)

which induces an isomorphism on homology (for algebras over a field). In [R] Rinehart defined the differential

B : C_{∗}(A, A) → C_{∗+1}(A, A)

which commutes with the Hochschild differential and becomes the de Rham differential under the isomorphism (0.1). A systematic study of a noncomutative de Rham theory, or cyclic homology, was undertaken in [C], [FT],[L],[T], etc. One defines the periodic cyclic complex (CC_{∗}^{per}(A), B + b) of an algebra A, and proves that when A is the ring of regular functions on a regular affine variety X over a field k of characteristic zero, the homology of this complex is the de Rham cohomology of X. It is worth mentioning that the correct space of “noncommutative differential forms” is not the Hochschild homology H_{∗}(A, A) but the space of all Hochschild chains C_{∗}(A, A). The Hochschild differential b itself is an odd noncommutative bivector field and the Hochschild homology is “a space of physical states” Kerb/Imb. We have already mentioned that there seems to be no way of recovering noncommutative differential forms together with their multiplicative structure. It turns out, however, that if one considers all differential operators on Ω^{∗}(X) (not just zero order multiplication operators) one gets a ring which has a noncommutative generalization. The construction we are going to present rests upon an idea of Yu.I.Manin ([M]).

Let R be a monoidal category. For an object A of R one calls an object \underline{End} A of R the inner object of endomorphisms of A if there are natural morphisms

\underline{End} A × \underline{End} A → \underline{End} A ; \underline{End} A × A → A  (0.5)

which are universal and associative in a natural way. In the category of associative algebras inner objects of endomorphisms do not exist (for example, the set of endomorphisms of an algebra is not a linear space). In this paper, however, we will show that, if one takes for \underline{End} A the differential graded algebra C^{∗}(A, A) of Hochschild cochains, the maps (0.5), in a sense, still exist. More precisely, they exist if one passes from the category of algebras to the category of complexes by means of some well known homological functors.

These functors put in correspondence to an algebra its Hochschild or periodic cyclic complex. In the case of bar complex the operations analogous to the ones that we will describe were discovered by Getzler - Jones and by Gerstenhaber - Voronov ([GJ], [GV]). Therefore one can say informally that in the homotopy category of (differential graded) algebras the inner endomorphism objects always exist.

The operations arising from the second map (0.5) (in the above sense) play a crucial role in our approach to index theorems (Section 6); they also allow to construct the bivariant JLO cochain and, hopefully, a Chern character with values in bivariant entire cyclic cohomology. Those operations bear a strong formal resemblance to the multiplication in the quantum cohomology of symplectic manifolds. In particular, one can construct “the Fukaya category” of an associative ring; objects of this category are automorphisms of the ring, and the object \underline{End} A from (0.5) is the object of endomorphisms of the identity.

The second author recalls Yu.I. Manin’s first reaction upon receiving A.Connes’ paper on noncommutative geometry in 1982. Commenting on Connes’ formula

da = i[F, a]

where F is an odd element, Prof. Manin suggested that perhaps the differentials should belong to the structure sheaf of a more general superscheme, not just of the odd affine line. This idea, together with I.M.Gelfand’s notion of an (𝔞, D)-system, is central to this work.

Let us describe the contents of the paper in more detail.

Let A be an associative unital algebra over a commutative unital ground ring k. Consider A as a bimodule over itself; by E_{A}^{∗} we denote the differential graded algebra (C^{∗}(A, A), δ, ⌣) which is the standard complex for computing Ext_{A⊗A^{∘}}^{∗}(A, A) = H^{∗}(A, A) (the Hochschild cohomology) equipped with the Yoneda product. We recall some well-known properties of this algebra in Section 1. In Section 2 we construct the maps of complexes

• : C_{∗}(A, A) ⊗ C_{∗}(E_{A}^{∗}, E_{A}^{∗}) → C_{∗}(A, A)
• : C_{∗}(E_{A}^{∗}, E_{A}^{∗}) ⊗ C_{∗}(E_{A}^{∗}, E_{A}^{∗}) → C_{∗}(E_{A}^{∗}, E_{A}^{∗})  (0.6)

Here C_{∗} stands for the Hochschild complex computing HH_{∗}(A) = Tor_{∗}^{A⊗A^{∘}}(A, A).

In Sect. 3 we construct similar morphisms

• : CC_{∗}^{per}(A) ⊗ CC_{∗}^{per}(E_{A}^{∗}) → CC_{∗}^{per}(A)
• : CC_{∗}^{per}(E_{A}^{∗}) ⊗ CC_{∗}^{per}(E_{A}^{∗}) → CC_{∗}^{per}(E_{A}^{∗})  (0.7)

where CC_{∗}^{per} is the periodic cyclic complex.

In particular, the center Z_{A} of A is a (differential) subalgebra of C^{∗}(A, A). One gets the usual products

C_{∗}(Z_{A}) ⊗ C_{∗}(A) → C_{∗}(A)
CC_{∗}^{per}(Z_{A}) ⊗ CC_{∗}^{per}(A) → CC_{∗}^{per}(A)

([CE], [HJ]).

On the other hand, let D be a Hochschild cochain of A (in particular a derivation). Put

i_{D}(a) = (−1)^{deg a·deg D}a • D

in the Hochschild complex where D, being an element of E_{A}^{∗}, is regarded as a zero-chain of E_{A}^{∗}; let

I_{D}(a) = (−1)^{deg a·deg D}a • D

but in the periodic cyclic complex; let

L_{D}(a) = (−1)^{deg a(deg D−1)}a • (1 ⊗ D)

(in this case the formulas are the same for Hochschild or periodic cyclic case). Then L_{D} is the Lie derivative and I_{D} is the contraction operator of Rinehart [R]; the fact that • is a morphism of complexes implies the Cartan homotopy formula

[B + b, I_{D}] = L_{D} + I_{δD}  (0.8)

(cf. [R]).

There is another product on C^{∗}(A, A), the Gerstenhaber bracket. It makes C^{∗}(A, A)[1] a differential graded Lie algebra which we denote by (𝔤(A), δ, [ , ]) (cf [Ge]). The Lie derivative L_{D} turns C_{∗}(A) and CC_{∗}^{per}(A) into differential graded 𝔤(A)-modules. Let R_{∗}(𝔤(A)) = ∧^{∗}(𝔤(A)) ⊗ U(𝔤(A)) be the standard Koszul resolution of the trivial right 𝔤(A)-module k. We construct a morphism of complexes of modules (Sect. 4):

J : R_{∗}(𝔤(A)) → \underline{End} CC_{∗}^{per}(A)  (0.9)

In our view, the existence of this morphism is one of the main features of noncommutative differential geometry. Compare this with the usual situation: let X be a smooth manifold, and let D_{i} be vector fields on X. Put

J(D_{1} ∧ ⋯ ∧ D_{m}) = i_{D_{1}} … i_{D_{m}} : Ω_{X}^{∗} → Ω_{X}^{∗−m}

Then

[d, J(D_{1} ∧ ⋯ ∧ D_{m})] = ∑_{i}(−1)^{m−i+1}J(⋯ ∧ D̂_{i} …)L_{D_{i}}+

+ ∑_{i<j}(−1)^{i+j−1}J([D_{i}D_{j}] ∧ ⋯ ∧ D̂_{i} ∧ ⋯ ∧ D̂_{j} …) (0.10)

The morphism (0.9) satisfies a similar formula in the general context (when, for example, D_{i} are derivations of A; of course, one has to replace d by B + b).

Note that one could simply put J(D_{1} ∧ ⋯ ∧ D_{m}) = I_{D_{1}} … I_{D_{m}} if the operators I_{D} (anti)-commuted; but they do not, so the formula is different.

In section 5 we construct another multiplication which is crucial for our approach to index theorems. Let A be an associative algebra. By \overline{C}_{∗−1}^{λ}(A) we denote the reduced cyclic complex of A; one can define this complex also as

\overline{C}_{∗−1}^{λ}(A) = Prim C_{∗}(𝔤𝔩(A), 𝔤𝔩(k); k)

(the subcomplex of primitive Lie algebra chains; cf. [L]). We construct the operation

\overline{C}_{∗−1}^{λ}(A) ⊗ CC_{∗}^{per}(A) ⟶ CC_{∗}^{per}(A)

or in other words the morphism of complexes

χ : \overline{C}_{∗−1}^{λ}(A) → \underline{End} CC_{∗}^{per}(A)

To obtain this operation one applies the homomorphism J in the case when A is replaced by its matrix algebra. Let M_{∞}(A) be the algebra of matrices over A with finitely many non-zero diagonals; let M(A) be its ideal of matrices with finitely many non-zero entries and let 𝔤𝔩 be the same algebra viewed as a Lie algebra. We will see in Section 5 that one can modify the operation J to get a homomorphism

C^{∗}(𝔤𝔩(A), 𝔤𝔩(k)) ⊗ CC_{∗}^{per}(M_{∞}(A)) → CC_{∗}^{per}(M(A))

Here 𝔤𝔩(A) is viewed as a subalgebra of the algebra of inner derivations. To get the operation χ one uses the embedding CC_{∗}^{per}(A) → CC_{∗}^{per}(M_{∞}(A)) and the trace map CC_{∗}^{per}(M(A)) → CC_{∗}^{per}(A).

The next four sections of the paper are devoted to examples and applications. We study the product • on the Hochschild complex. One can define the Hochschild chain complex of the algebra of cochains in two different ways:

\begin{matrix}C^{n}(A) = ⨁_{i+j=n} C_{−i}(E_{A}^{∗}, E_{A}^{∗})^{j} \\ C_{∞}^{n}(A) = ∏_{i+j=n} C_{−i}(E_{A}^{∗}, E_{A}^{∗})^{j} \end{matrix}

Therefore we get two versions of the cohomology ring, H^{∗}(A) and H_{∞}^{∗}(A), together with a map H^{∗} → H_{∞}^{∗}. First let A = k[X], where X is regular affine over a ring of characteristic zero. We show that

H^{∗}(A) = D(Ω_{X}^{∗})^{o}; H_{∞}^{∗}(A) = End(Ω_{X}^{∗})

Here D(Ω_{X}^{∗}) stands for the ring of differential operators on the graded module of differential forms, that is to say, differential operators on the exterior algebra of the cotangent bundle; the symbol ^{o} means the opposite ring).

Next we consider the case of a deformed algebra of functions. Let (M, ω) be a symplectic manifold and ∗ be a star product on C^{∞}(M) (cf. [BFFLS]) such that f ∗ g − g ∗ f = iℏ{f, g} + O(ℏ^{2}). Isomorphism classes of such star products are parametrized by H^{2}(M, ℂ[[ℏ]]) ([LDW], [De], [NT]). Let A = 𝔸^{ℏ}(M)[ℏ^{−1}] = C^{∞}(M)[[ℏ, ℏ^{−1}] with the product ∗. Then

H^{∗}(A) ˜→ H^{∗}(M^{S^{1}}, ℂ[[ℏ, ℏ^{−1}])

if M is simply connected;

H_{∞}^{∗}(A) ˜→ H^{∗}(M, ℂ[[ℏ, ℏ^{−1}])

Note some resemblance with the multiplication on Floer cohomology.

Let us now mention some applications of our constructions.

In Section 6 we outline the proof of the algebraic index theorem from [Fe] and [NT]. Our approach owes very much to the ideas of B.Feigin and I.M.Gelfand. Let us mention some other applications, namely to deformations and characteristic classes.

Let ∇ be an odd element of 𝔤(A) such that

δ∇ + \frac{1}{2}[∇∇] = 0.  (0.11)

(δ is the Hochschild differential). We show that, \underline{formally}, i.e. without regard for convergence, if

e^{∇} = ∑_{n≥0} \frac{∧^{n}∇}{n!} ⊗ 1  (0.12)

in ∧^{∗}𝔤 ⊗ U𝔤, then for X(∇) = J(e^{∇} ⊗ 1)

[B + b, X(∇)] = L_{∇}X(∇)  (0.13)

Assume, for example, that A is a ℤ_{2}-graded algebra and D / is an odd element of A. Then

∇ = ad(D /) − D /^{2}

is an odd cochain satisfying (0.11). Using (0.9), we construct a morphism

ch(D /) : CC_{∗}^{per,(0)}(A) → CC_{∗}^{per}(A)

where CC_{∗}^{per,(0)} is the subcomplex of finite cochains:

CC_{n}^{per,(0)}(A) = ⨁_{i≡n(2)}A ⊗ \overline{A}^{⊗n},

while, as usually,

CC_{n}^{per}(A) = ∏_{i≡n(2)}A ⊗ \overline{A}^{⊗n}.

The component

ch_{0}(D /) : CC_{∗}^{per,(0)}(A) → A

is given by the formula from [JLO], [GS]:

ch_{0}(D /) = ∑_{n≥0} ∫_{\begin{matrix}t_{0}+···+t_{n}=1 \\ t_{i}≥0 \end{matrix}}a_{0}e^{−t_{0}D̸^{2}}[D / , a_{0}] … e^{−t_{n−1}D̸^{2}}[D / , a_{n}]e^{−t_{n}D̸^{2}}dt_{0} … dt_{n−1}

Another application: let A be an algebra; assume that another multiplication law on A, ∗, is given. Put

∇(a, b) = a ∗ b − ab;

then ∇ satisfies (0.11). The operator X(∇) is a morphism of complexes

X(∇) : CC_{∗}^{per,(0)}(A, ·) → CC_{∗}^{per}(A, ∗)

When ab − a ∗ b ∈ I where I is an ideal of A such that A is I-adically complete, then X(∇) gives a well-defined isomorphism of periodic cyclic complexes, an explicit version of the theorem of Goodwillie. If one composes this isomorphism with a trace on the algebra (A, ∗), one recovers the cocycle of [CFS]. One can expect that, if the multiplications · and ∗ are “close”, one would be able to use X(∇) to compare Connes’ entire cyclic cohomologies.

In Section 10 we try to clarify the analogy between the multiplication • and the quantum multiplication, as well as to advance B.Feigin’s idea that Lagrangian intersections might be related to the cohomology of deformed algebras. We construct “the Fukaya category” of an associative algebra A; the objects of this category are automorphisms of A and, for two endomorphisms α and β, the complex Hom(α, β) is the twisted cochain complex C_{∗}(E_{A}^{∗},_{α} E_{β}^{∗}). We also construct a functor putting in correspondence to α the twisted chain complex C_{∗}(A, A_{α}). Conjecturally, these are an A_{∞} category and an A_{∞} functor. When A is a deformed ring of functions on a symplectic manifold then the cohomology of the above complexes is related to their fixed point sets and to the cohomology of certain path and loop spaces.

\underline{Acknowledgements} The second author has many fond memories of the time when he was Yu. I. Manin’s graduate student in Moscow from 1982 to 1985. Some of the ideas and methods of this paper are due to his collaboration with Yu.L. Daletski, B. Feigin, I.M. Gelfand, as well as to discussions with Yu. I. Manin, A. Beilinson, V. Schechtman, M. Wodzicki and many others back then (and later); some other methods and ideas came from the first author’s collaboration with G. Elliot and T. Natsume. We would like to thank P. Bressler, J.L. Brylinski, A.Connes, B.V. Fedosov, M. Flato, E.Getzler, J. Kaminker, C. Kassel, M. Kontsevich, J.L. Loday, A.Radul, N.Reshetikhin, D. Sternheimer, D.Tamarkin and A. Voronov for fruitful discussions. Note that homological operations somewhat similar to ours were first considered by Getzler in [G]; there, too, a homological complex of the differential graded algebra (C^{∗}(A, A), δ, ⌣) appeared. Finally, we would like to thank the referee whose many suggestions greatly improved the presentation.

Section 1.

Hochschild cohomological complex.

Let A be a graded algebra with unit over a commutative unital ring k. Let M be a graded bimodule over A. A Hochschild d-cochain is a linear map A^{⊗d} → M. Put, for d ≥ 0,

C^{d}(A, M) = Hom_{k}(\overline{A}^{⊗d}, M)

where \overline{A} = A/k · 1. For a cochain D in C^{∗}(A, A) put

deg D = (degree of the linear map D) + d
|D| = deg D − 1; |a| = deg a − 1

Given a tensor a_{1} ⊗ ⋯ ⊗ a_{N} in A^{⊗N}, we will denote it by (a_{1}, …, a_{n}). We will write η_{j} = j ; ∑_{i=1}|a_{i}| (as in [G]). Put for cochains D and E in C^{∗}(A, A),

\begin{matrix}(D ⌣ E)(a_{1}, …, a_{d+e}) = (−1)^{deg E·η_{d}}D(a_{1}, …, a_{d}) \\ ×E(a_{d+1}, …, a_{d+e}); \end{matrix}  (1.1)

\begin{matrix}(D ∘ E)(a_{1}, …, a_{d+e−1}) = ∑_{j≥0}(−1)^{|E|η_{j}}D(a_{1}, …, a_{j}, \\ E(a_{j+1}, …, a_{j+e}), …); \\ [D, E] = D ∘ E − (−1)^{|D||E|}E ∘ D \end{matrix}  (1.2)

These operations define the graded associative algebra (C^{∗}(A, A), deg , ⌣) and the graded Lie algebra (C^{∗}(A, A), | · |, [ , ]) (cf. [CE]; [Ge]). Let

m(a_{1}, a_{2}) = (−1)^{deg a_{1}} a_{1}a_{2};

this is a 2-cochain of A (not in C^{2}, because it is a bilinear map from A to A, not from \overline{A} to A). Put

\begin{matrix}δD = [m, D]; \\ (δD)(a_{1}, …, a_{d+1}) = (−1)^{|a_{1}||D|+|a_{1}|+1}× \\ × a_{1}D(a_{2}, …, a_{d+1})+ \\ + d ; ∑_{j=1}(−1)^{|D|+η_{j}}D(a_{1}, …, a_{j} a_{j+1}, …, a_{d+1}) \\ + (−1)^{|D|+η_{d}+1}D(a_{1}, …, a_{d})a_{d+1} \end{matrix}  (1.3)

The last formula defines a differential of degree +1 on C^{∗}(A, M) for any M. For an element x of A, let \underline{x} be the corresponding zero-cochain in C^{∗}(A, A). By definition

(δ\underline{x})(a) = (−1)^{deg x}[x, a];

a one-cochain D is a cocycle iff it is a derivation. One has

\begin{matrix}δ^{2} = 0; δ(D ⌣ E) = δD ⌣ E + (−1)^{deg D}D ⌣ δE \\ δ[D, E] = [δD, E] + (−1)^{|D|} [D, δE] \end{matrix}

(δ^{2} = 0 follows from [m, m] = 0).

Thus C^{∗}(A, A) becomes a complex; the cohomology of this complex is H^{∗}(A, A) or the Hochschild cohomology. The ⌣ product induces the Yoneda product on H^{∗}(A, A) = Ext_{A⊗A^{0}}^{∗}(A, A). The operation [ , ] is the Gerstenhaber bracket [Ge].

If (A, ∂) is a differential graded algebra then one can define the differential ∂ acting on C^{∗}(A, A) by

∂D = [∂, D]

\underline{Definition 1.1}. Define the differential graded algebra E_{A}^{∗} as the complex C^{∗}(A, A) with the differential δ (or δ +∂ if (A, ∂) is a differential graded algebra), the grading deg and the product ⌣ .

In Sections 1–9 the only cochains we will be considering will be those from C^{∗}(A, A), i.e. when A = M. For Hochschild cochains D_{i} define a new Hochschild cochain D_{0}{D_{1}, …, D_{m}} by the following formula of Gerstenhaber ([Ge]) and Getzler ([G]):

D_{0}{D_{1}, …, D_{m}}(a_{1}, …, a_{n}) =

= ∑(−1)^{∑_{p=1}^{m} η_{ip} |D_{p}|}D_{0}(a_{1}, …, a_{i_{1}}, D_{1}(a_{i_{1}+1}, …), …, D_{m}(a_{i_{m}+1}, …), …)

(1.4)

The following statements are contained in [Ge], [G], [GV]. \underline{Proposition 1.2}. One has

\begin{matrix}(D\{E_{1}, …, E_{k}\})\{F_{1}, …, F_{l}\} = ∑(−1)^{∑_{q≤ip} |E_{p}||F_{q}|}× \\ ×D\{F_{1}, …, E_{1}\{F_{i_{1}+1}, …, \}, …, E_{k}\{F_{i_{k}+1}, …, \}, …, \} \end{matrix}

\underline{Proof}. Direct computation ■ \underline{Example 1.3}. One has

D ⌣ E = (−1)^{deg D}m{D, E}; D ∘ E = D{E}

\underline{Corollary 1.4}.

1). (D ∘ E) ∘ F − D ∘ (E ∘ F) = D{E, F } + (−1)^{|E||F |}D{F, E};

(−1)^{|D||F |}[[D, E], F] + (−1)^{|E||D|}[[E, F], D] + (−1)^{|F ||E|}[[F, D], E] = 0 .

2). Let M be the cochain defined above, but for the differential graded algebra E_{A}^{∗}: M(D_{1}, D_{2}) = (−1)^{deg D_{1}} D_{1} ⌣ D_{2}. For any cochain D in C^{∗}(A, A) define a cochain \overline{D} in C^{∗}(A, A) by \overline{D}(E_{1}, …, E_{k}) = D\{E_{1}, …, E_{k}\} for all k. Then

i).

\overline{(D ⌣ E)} = \overline{D} ⌣ \overline{E};

ii).

\overline{δD} = [δ, \overline{D}] + [M, \overline{D}].

Proof. To prove the first identity in 1), one applies Proposition 1.2 in the case when k = l = 1. The second identity of 1) follows from the first if one antisymmetrizes the first (with appropriate signs). To prove the identity 2), i) one applies Proposition 1.2 to compute (m{D, E}){F_{1}, …, F_{m}} to get the following formula

(D ⌣ E){F_{1}, …, F_{m}} = ∑_{p=0}^{m}(−1)^{deg E ∑_{i≤p} |F_{i}|}D{F_{1}, …, F_{p}} ⌣ E{F_{p+1}, …, F_{m}} .

which is equivalent to 2), i).

To prove 2), ii) one applies Proposition 2.2 to compute (m{D}){F_{1}, …, F_{m}}, (D{m}){F_{1}, …, F_{m}}, and (D{F_{1}, …, F_{m}}){m}; their sum, with appropriate sings, gives the following identity

δD({F_{1}, …, F_{m}}) = (δD){F_{1}, …, F_{m}}−

− ∑_{p=1}^{m}(−1)^{degD+∑_{i<p} |F_{i}|}D{F_{1}, …, δF_{p}, …, F_{m}}+

+ ∑_{p=1}^{m−1}(−1)^{degD+∑_{i≤p} |F_{i}|}D{F_{1}, …, F_{p} ⌣ F_{p+1} …, F_{m}}−

−(−1)^{degD+∑_{i<m} |F_{i}|}D{F_{1}, …, F_{m−1}} ⌣ F_{m}+
+(−1)^{degD|F_{1}|}F_{1} ⌣ D{F_{2}, …, F_{m}}

which is equivalent to 2), ii).

Section 2.

Operations on the Hochschild Complex.

Let M be a differential graded bimodule over a differential graded algebra A. Recall that the Hochschild homological complex (C_{∗}(A, M), b) is the following:

C_{n}(A, M) = M ⊗ \overline{A}^{⊗n}

We will always write, as in Section 1,

(a_{0}, …, a_{n}) = a_{0} ⊗ ⋯ ⊗ a_{n}

For a_{0} in M and a_{i} in A, i > 0, define

b(a_{0}, …, a_{n}) = n−1 ; ∑_{j=0}(−1)^{η_{j+1}+1}(a_{0}, …, a_{j} a_{j+1}, …, a_{n})

+ (−1)^{(|a_{n}|+1)(η_{n}+1)+1}(a_{n}a_{0}, a_{1}, …, a_{n−1}). (2.1)

We shall denote C_{∗}(A, A) simply by C_{∗}(A). We introduce a grading on C_{∗}(A, A) by the formula

deg (a_{0}, …, a_{n}) = ∑ dega_{i} + n

(A rule for remembering the signs: let |ϵ| = 1, ϵ^{2} = 0; map C_{n}(A) to A[ϵ]/[A[ϵ], A[ϵ]], a_{0} ⊗ ⋯ ⊗ a_{n} ↦ a_{0}ϵa_{1}ϵ ⋯ a_{n}ϵ; then b becomes \frac{∂}{∂ϵ}).

The homology of C_{∗}(A, M) is the Hochschild homology HH_{∗}(A, M) = Tor_{∗}^{A⊗A^{∘}}(A, M). We will denote HH_{∗}(A, A) simply by HH_{∗}(A). If A is a differential graded algebra and ∂ the differential in A, one extends ∂ to C_{∗}(A):

∂(a_{0}, …, a_{n}) = n ; ∑_{j=0}(−1)^{η_{j}}(a_{0}, …, ∂a_{j}, …, a_{n})  (2.2)

For a in C_{∗}(A, A) and x in C_{∗}(E_{A}^{∗}, E_{A}^{∗}) define

a • x = a •_{1} x + a •_{2} x  (2.3)

where

(a_{0}, …, a_{n}) •_{1} (D_{0}, …, D_{m}) = ∑(−1)^{∑_{i>0} |a_{i}|degD_{0}+∑_{p>0} ∑_{i>ip} |a_{i}||D_{p}|}× × (a_{0}D_{0}(a_{1}, a_{2}, …), …, a_{i_{1}}, D_{1}(a_{i_{1}+1}, …), …, D_{m}(a_{i_{m}+1}, …), …) (2.4)

(a_{0}, …, a_{n}) •_{2} (D_{0}, …, D_{m}) =

= ∑_{q≤n+1}(−1)^{∑_{i<q} |a_{i}| ∑_{i≥q} |a_{i}|+∑_{i_{0}<i<q} |a_{i}|degD_{0}+∑_{p>0} ∑_{ip<i<q} |a_{i}||D_{p}|}×

× (D_{m}(a_{q}, …, a_{n}, a_{0}, …, a_{i_{0}})D_{0}(a_{i_{0}+1}, …), …, a_{i_{1}}, , D_{1}(a_{i_{1}+1}, …), …, D_{m−1}(a_{i_{m−1}+1}, …), …) (2.5)

The sum in (2.5) is taken over all q, i_{0}, …, i_{m−1} for which a_{0} is inside D_{m}. \underline{Theorem 2.1}. The map

• : C_{∗}(A, A) ⊗ C_{∗}(E_{A}^{∗}, E_{A}^{∗}) → C_{∗}(A, A)  (2.6)

is a morphism of complexes, i.e.

b(a • x) = (ba) • x + (−1)^{deg a}a • (b + δ)x

\underline{Proof}. Let α = (a_{0}, …, a_{n}) and D = (D_{0}, …, D_{m}). Compute b(α•_{1}D)−(bα)•_{1}D. It will consist of the following terms.

1). All the terms with a_{i}a_{i+1} or a_{n}a_{0} outside D_{j}, i > 0. They will cancel out (because they appear once in b(α •_{1} D) and once in (bα) •_{1} D).

1a).(a_{0}a_{1}D_{0}(…), … D_{1}(…), …) 2). All the terms with a_{i}a_{i+1} or a_{n}a_{0} inside D_{j}, i > 0. 3). All the terms with a_{i}D_{j}(…) or D_{j}(…)a_{i}, i > 0. The terms 1a), 2), 3) will cancel out with the term α •_{1} δD. 4).(a_{0}D_{0}(…)D_{1}(…), …, D_{k}(…), …);

(a_{0}D_{0}(…), …, D_{k}(…)D_{k+1}(…), …);

These terms will cancel out with all the terms in α •_{1} bD except the following:

(a_{0}D_{m}(…)D_{0}(…), …, D_{1}(…), …)  (2.7)

5). (D_{m}(…)a_{0}D_{0}(…), …, D_{1}(…), …).

Now compute b(α •_{2} D) − (bα) •_{2} D. It will consist of the following terms. 6). All the terms with a_{i}a_{i+1} or a_{n}a_{0} outside D_{j}. They will cancel out (because

they appear once in b(α •_{2} D) and once in (bα) •_{2} D).

7). All the terms with a_{i}a_{i+1} or a_{n}a_{0} inside D_{j}. 8). All the terms with a_{i}D_{j}(…) or D_{j}(…)a_{i}. The terms 7), 8) will cancel out with some of the terms in α •_{2} δD 9).D_{m}(… a_{0} …)D_{0}(…), …, D_{1}(…), …). 9a). D_{m}(… a_{0} …)D_{0}(…)D_{1}(…), …, D_{2}(…), …). 9b).D_{m}(… a_{0} …)D_{0}(…), …, D_{k}(…)D_{k+1}(…), …). The terms 9) and 9a) cancel out with all the terms in α•_{2}bD except the following:

((D_{m−1} ⌣ D_{m})(…, a_{0}, …)D_{0}(…), …, D_{1}(…), …)  (2.8)

and

D_{m−1}(…, a_{0}, …)D_{m}(…)D_{0}(…), …, D_{1}(…), …)  (2.9)

The combination of the terms 9b), (2.8) and (2.9) cancels out. The remaining terms are (2.7) and 5). But they will cancel out with the terms in α •_{2} δD that do not cancel out with 7), 8).

■

Consider some examples of the product •. When all D_{i} are zero-cochains then one gets the shuffle product from [CE]. On the other hand, put, for a d-cochain D and for a in C_{∗}(A, A), i_{D}(a) = (−1)^{deg a deg D}(a • D);

i_{D}(a_{0}, …, a_{n}) = (−1)^{deg a_{0}·deg D_{0}}(a_{0}D(a_{1}, …, a_{d}), a_{d+1}, …, a_{n})  (2.7)

One gets the cap product from [CE]:

C^{d}(A, A) ⊗ C_{n}(A, A) → C_{n−d}(A, A)

Now, for a d-cochain D and for a in C_{∗}(A, A), let L_{D}(a) = (−1)^{|D|·deg a}a • (1, D); one has

L_{D}(a_{0}, …, a_{n}) = ∑_{q=0}^{n−d+2}(−1)^{(η_{n+1}−η_{q})η_{q}+|D|}(D(a_{q}, …, a_{0}, …), …, a_{q−1})+

+ ∑_{k=1}^{n−d}(−1)^{(η_{k+1}+1)|D|}(a_{0}, …, a_{k}, D(a_{k+1}, …), …,) (2.10)

One has

[b, L_{D}] = −L_{δD}

Now construct the product

• : C_{∗}(E_{A}^{∗}, E_{A}^{∗}) ⊗ C_{∗}(E_{A}^{∗}, E_{A}^{∗}) → C_{∗}(E_{A}^{∗}, E_{A}^{∗})  (2.11)

as follows. For a cochain D let D^{(k)} be the following k-cochain of E_{A}^{∗}:

D^{(k)}(D_{1}, …, D_{k}) = D{D_{1}, …, D_{k}}

\underline{Proposition 2.2}. The map

D ↦ ∑_{k≥0} D^{(k)}

is a morphism of differential graded algebras E_{A}^{∗} → E_{E_{A}^{∗}}^{∗}. \underline{Proof}. Follows from Corollary 1.4, 2), 3). ■

Now we construct the product • by composing the one for the algebra E_{A}^{∗} with the above map.

The product (2.11) is given by formulas (2.3-2.5) where a_{i} are now viewed as Hochschild cochains and D(a_{1}, …, a_{d}) is replaced by D{a_{1}, …, a_{d}}. \underline{Proposition 2.3}. The product (2.11) is homotopically associative.

We omit the proof. Section 3.

Operations on the periodic cyclic complex.

In the situation of Sect. 2 define

B(a_{0}, …, a_{n}) = n ; ∑_{j=0}(−1)^{(η_{n+1}−η_{j})η_{j}}(1, a_{j}, …, a_{n}, a_{0}, …, a_{j−1})  (3.1)

Then B^{2} = b^{2} = Bb + bB = 0; denote

CC_{n}^{per}(A) = ∏_{i≡n(2)}A ⊗ \overline{A}^{⊗n}  (3.2)

with the differential b + B + ∂ (for the differential graded algebra (A, ∂)). Put

(a_{0}, …, a_{n}) •_{3} (D_{1}, …, D_{m}) =

= ∑_{0≤p≤m}(−1)^{∑_{r<p}(|D_{r}|+1) ∑_{r≥p}(|D_{r}|+1)+∑_{r}|D_{r}|}×

× ∑_{0≤j≤n}(−1)^{(η_{n+1}−η_{j})η_{j}}λ′(D_{p}, …, D_{m}, D_{0}, …, D_{p−1})(1, a_{j}, …, a_{n}, a_{0}, …, a_{j−1}) (3.3)

Here λ(D_{1}, …, D_{m})(a_{0}, …, a_{n}) =

= ∑_{j_{1}>0}(−1)^{|D_{1}|(η_{n+1}−η_{j_{1}} )+···+|D_{m}|(η_{n+1}−η_{jm})}×

× (a_{0}, …, D_{1}(a_{j_{1}} …), …, D_{m}(a_{j_{m}}, …), …); (3.4)

λ′ is the sum of all those terms in λ which contain D_{0}(…) after a_{0}. \underline{Theorem 3.1}. The map

• : CC_{∗}^{per}(A) ⊗ CC_{∗}^{per}(E_{A}^{∗}) → CC_{∗}^{per}(A)  (3.5)

is a morphism of complexes. \underline{Proof}. Let α = (a_{0}, …, a_{n}) and D = (D_{0}, …, D_{m}). Compute b(α •_{3} D) − (bα) •_{3} D − α •_{3} bD. It will consist of the following terms.

1). All the terms with a_{i}a_{i+1} or a_{n}a_{0} outside D_{j}. They will cancel out (because they appear once in b(α •_{3} D) and once in (bα) •_{3} D).

2). All the terms with a_{i}a_{i+1} or a_{n}a_{0} inside D_{j}.

3). All the terms with a_{i}D_{j}(…) or D_{j}(…)a_{i}.

The terms 2), 3) except for the following one (formula (3.6) will cancel out with the term α •_{3} δD.

(1, …, a_{0}D_{0}(…), …, D_{1}(…), …)  (3.6)

4). All the terms with D_{i}(…)D_{i+1}(…) or D_{m}(…)D_{0}(…) but for the following (3.7) will cancel out.

(1, …, D_{m}(…, a_{0}, …)(D_{0}(…), …, D_{1}(…), …)  (3.7)

5).(D_{k}(…, a_{0},_{l} dots), …, D_{0}(…), …).

6).(a_{0}, …, D_{0}(…), …).

7). D_{0}(…), …, a_{0}, …, D_{m}(…), …).

(Here a_{0} may be inside or outside D_{i}).

8). (a_{k}, …, a_{0}, …, D_{0}(…], …)

where a_{0} may be inside or outside D_{i}. These terms will cancel out because they will enter twice (namely, in the first and in the last summands of b(α •_{3} D).

The term 5) will cancel out with α •_{2} BD, the term 6) with α •_{1} BD, and the term 7) with Bα •_{1} D. The term (3.6) cancels out with B(α •_{1} D) and the term (3.7) with B(α •_{2} D). Now, Bα •_{2} D = 0 and Bα •_{3} D = α •_{3} BD = B(α •_{3} D) = 0, which finishes the proof. ■

For a d-cochain D and for a in CC_{∗}^{per}(A), let I_{D}(a) = (−1)^{deg D·deg a}(a • D); L_{D}(a) = (−1)^{|D|·deg a}a • (1, D); one has

[B + b, I_{D}] − I_{δD} = L_{D}.
■

This is the homotopy formula of Rinehart [R].

One can define the product

• : CC_{∗}^{per}(E_{A}^{∗}) ⊗ CC_{∗}^{per}(E_{A}^{∗}) → CC_{∗}^{per}(E_{A}^{∗})  (3.8)

exactly in the same way as in the end of Section 2.

\underline{Remark 3.2}. Considering CC_{∗}^{per}(A) = CC_{∗}^{per}(E_{A}^{0}) as a part of CC_{∗}^{per}(E_{A}^{∗}) and restricting the product • to it, we get the Hood-Jones product on CC_{∗}^{per}(A). When A is commutative then CC_{∗}^{per}(A) is a subcomplex of CC_{∗}^{per}(E_{A}^{∗}) and one gets a product on the complex CC_{∗}^{per}(A).

Section 4.

The Lie algebra complex.

Let (𝔤, ∂) be a differential graded Lie algebra. Put ∧^{∗}𝔤 = T (𝔤)/ < D ⊗ E − (−1)^{(|D|−1)(|E|−1)}E ⊗ D >;

R_{∗}(𝔤) = ∧^{∗}𝔤 ⊗ U(𝔤);

∂^{Lie}(D_{1} ∧ ⋯ ∧ D_{m} ⊗ f) = ∑_{i}(−1)^{θ_{i}}D_{1} ∧ …

∧D̂_{i} ∧ ⋯ ∧ D_{m} ⊗ D_{i}f+

+∑_{i<j}(−1)^{θ_{ij}}D_{1} ∧ ⋯ ∧ D̂_{i} ∧ ⋯ ∧ [D_{i}D_{j}] ∧ ⋯ ⊗ f

where

θ_{i} = ∑_{r<i}(|D_{r}| + 1) + |D_{i}|∑_{r>i}(|D_{r}| + 1);
θ_{ij} = ∑_{r<i}(|D_{r}| + 1) + |D_{i}|( ∑_{i<r<j}(|D_{r}| + 1) + 1);

Put also

∂(D_{1} ∧ ⋯ ∧ D_{m}) = ∑(−1)^{μ_{j}}(D_{1}, …, δD_{j}, …, D_{m})

where

μ_{j} = 1 + ∑_{r<j}(|D_{r}| + 1).

\underline{A rule for remembering the signs}: Let |ϵ| = 1, ϵ^{2} = 0; map R_{∗}(𝔤) to U(𝔤[ϵ]) by (D_{1} ∧ ⋯ ∧ D_{m}) ⊗ f ↦ ϵD_{1} … ϵD_{m} · f; then ∂^{Lie} becomes \frac{∂}{∂ϵ}; ∂ is the induced derivation on U but with the opposite sign.

Recall that in Section 2 we defined Lie derivatives L_{D} : CC_{∗}^{per}(A) ↦ CC_{∗}^{per}(A). \underline{Lemma 4.1}.

[L_{D}, L_{E}] = L_{[D,E]}

\underline{Proof}. Straightforward. ■ Put 𝔤(A) = (C^{∗}(A, A); | · |; [ , ]; δ). One sees that 𝔤(A) is a differential graded

Lie algebra and that U(𝔤(A)) acts on the complex of endomorphisms \underline{End} CC_{∗}^{per}(A) from the right: X ↦ X · L_{D} for D ∈ 𝔤(A). For f ∈ U(𝔤(A)), we will denote the corresponding operator by L_{f}.

The purpose of this Section is to show that the complex R_{∗}(𝔤(A)) acts 𝔤(A) - equivariantly on the complex CC_{∗}^{per}(A) , thus generalizing the standard calculus from the commutative case.

Let us consider the complex CC_{∗}^{per}(E_{A}^{∗}) with the Hood-Jones product (Remark 3.2). In other words, we view E_{A}^{∗} as C_{0}(E_{A}^{∗}) and consider the product • from section 3. We denote this product by ⋆. \underline{Definition 4.2}. Put

J(D_{1} ∧ ⋯ ∧ D_{m})α = (−1)^{deg α ∑_{i=1}^{m} degD_{i}} \frac{1}{m!}∑_{σ}ϵ_{σ}α • (D_{σ_{1}} ⋆ (D_{σ_{2}} ⋆ (… ⋆ D_{σ_{m}})) …)

where the sign ϵ_{σ} is taken with respect to the rule

ϵ_{(i,i+1)} = (−1)^{(|D_{i}|−1)(|D_{i+1}|−1)}

If f is in U(𝔤(A)) put

J(D_{1} ∧ ⋯ ∧ D_{m} ⊗ f) = J(D_{1} ∧ ⋯ ∧ D_{m})L_{f}.

\underline{Theorem 4.3}.

i) The map J : R_{∗}(𝔤(A)) → \underline{End} (CC_{∗}^{per}(A)) is a homomorphism of right 𝔤(A)-modules;

ii)

[B + b, J(γ)] = J((∂^{Lie} + δ)γ)

for γ in R(𝔤(A); iii) If E_{1}, …, E_{n}, D are in E^{0} or in E^{1} then

[L_{D}, J(E_{1} ∧ … ∧ E_{n})] = ∑_{j=1}^{n}(−1)^{|D| ∑_{i<j}(|E_{i}|+1)}J(E_{1} ∧ … [D, E_{j}] ∧ … E_{n})

\underline{Proof}. i) is obvious. To prove ii) and iii), one needs the following partial associativity properties of the product •. \underline{Lemma 4.4.}. Let α ∈ CC_{∗}^{per}(A), β ∈ CC_{∗}^{per}(E_{A}^{∗}) and γ ∈ E_{A}^{∗}. Then

i) α • ((1, D) • β) = (α • (1, D)) • β;

ii) (α • β) • (1, D) = α • (β • (1, D)) if D is in E^{0} or in E^{1}; iii)

α • (1, D) • β − (−1)^{|β||D|}α • β • (1, D) = α • [D, β]

where D is in E^{0} or in E^{1} and [D, β] stands for the usual action of 𝔤(A) on its tensor powers;

iv)

\begin{matrix}(1, D) • (E_{0}, …, E_{n}) = (1, D) ⋆ (E_{0}, …, E_{n})+ \\ + ∑_{j=0}^{n}(−1)^{|D|(∑_{i≤j} |E_{i}|+1)}(E_{0}, …, E_{j} ∘ D, …, E_{n}) \end{matrix}

v) D ⋆ ((1, E) ⋆ γ) = (−1)^{(|D|−1)|E|}(1, E) ⋆ (D ⋆ γ)

\underline{Proof.}. Direct computation. ■

Now let us prove the theorem. We shall omit the signs for simplicity. One has [B + b, J(D_{1} ∧ ⋯ ∧ D_{m})]α = α • Alt(B + b + δ)((D_{1} ⋆ (D_{2} ⋆ (… ⋆ D_{m})) …)) =

= α • (Alt ∑_{j} ±(D_{1} ⋆ … ((1, D_{j}) ⋆ (… ⋆ D_{m})) …)+

∑_{j} ±(D_{1} ⋆ … (ad(D_{j}) ⋆ (… ⋆ D_{m})) …)+

+ ∑_{j} ±(D_{1} ⋆ … (δ(D_{j}) ⋆ (… ⋆ D_{m})) …))

The second sum contributes terms D_{i} ⌣ D_{j} − (−1)^{(|D_{i}|−1)(|D_{j}|−1)}D_{j} ⌣ D_{i} and therefore vanishes under antisymmetrization. The third sum contributes the term J(δγ)α. The first sum, because of (v), is equal to

α • Alt ∑_{j}(1, D_{j}) ⋆ (±(D_{1} ⋆ … (D̂_{j} ⋆ (… ⋆ D_{m})) …) =

= α • Alt ∑_{j} ±(1, D_{j}) • (D_{1} ⋆ … (D̂_{j} ⋆ (… ⋆ D_{m})) …)+

+ α • Alt ∑_{j} ±(D_{1} ⋆ … (D_{i} ∘ D_{j} ⋆ (… (D̂_{j} ⋆ (… ⋆ D_{m})) …)+

= ∑_{j} ±J(D_{1} ∧ … ∧ D̂_{j} ∧ … ∧ D_{m})(L_{D}α)+

+ ∑_{i<j} ±J([D_{i}, D_{j}] ∧ … ∧ D̂_{i} ∧ … D̂_{j} ∧ … ∧ D_{m})(α)

■

Section 5.

The characteristic map χ In this section we construct a natural morphism of complexes

\overline{C}_{∗−1}^{λ}(A) ⊗ CC_{∗}^{per}(A) → CC_{∗}^{per}(A)

where \overline{C}_{∗−1}^{λ}(A) is the reduced cyclic complex equipped with the differential b: \overline{C}_{n}^{λ}(A) = \overline{A}^{⊗(n+1)}/Im(1 − τ) where \overline{A} = A/k · 1 and

τ(a_{0}, …, a_{n}) = (−1)^{η_{n}(η_{n}−η_{n−1})}(a_{n}, a_{0}, …, a_{n−1}).

Note that this complex is well defined ([L]).

Let M(A) be the algebra of matrices (a_{ij})_{1≤i,j≤∞} for which a_{ij} ∈ A and all but finitely many of a_{ij} are zero. The same space considered as a Lie algebra is 𝔤𝔩(A). The formula

(a_{0}, … a_{n}) ↦ (−1)^{∑ deg a_{i}}E_{01}^{a_{0}} ∧ E_{12}^{a_{1}} ∧ ⋯ ∧ E_{n0}^{a_{n}}

defines a morphism of complexes

\overline{C}_{∗−1}^{λ}(A) → C_{∗}(𝔤𝔩(A), 𝔤𝔩(k); k)

(with the relative Lie algebra chains in the right hand side). This map is an isomorphism of \overline{C}_{∗}^{λ}(A) with the subcomplex of primitive elements; cf. [T],[FT], [LQ].

Let M_{∞}(A) be the associative algebra of matrices (a_{ij})_{1≤i,j≤∞} such that a_{ij} ∈ A and a_{ij} = 0 on all but finitely many diagonals. Consider the differential graded Lie subalgebra C^{≤1}(M(A), M(A)). There is a morphism to this algebra from the differential graded Lie algebra (𝔤𝔩(A[η]) where η is a formal parameter of degree −1 such that η^{2} = 0; the differential on A[η] is ∂/∂η. This morphism sends ηa to a in C^{0} and a to ad(a) in C^{1}. From Theorem 4.3 one gets the pairing

R_{∗}(𝔤𝔩(A[η])) ⊗ CC_{∗}^{per}(M_{∞}(A)) ⟶ CC_{∗}^{per}(M(A))  (5.1)

Consider the following morphisms:

CC_{∗}^{per}(A) ⟶^{i} CC_{∗}^{per}(M_{∞}(A));

CC_{∗}^{per}(M(A)) ⟶^{tr} CC_{∗}^{per}(A)

The map i is induced by the inclusion a ↦ a · 1; the second map tr acts as follows:

(a_{0}m_{0}, …, a_{n}m_{n}) ⟼ tr(m_{0} … m_{n}) (a_{0}, …, a_{n})

for a_{i} ∈ A and m_{i} ∈ M(k). Composing (5.1) with these maps we obtain

R_{∗}(𝔤𝔩(A[η])) ⊗ CC_{∗}^{per}(M_{∞}(A)) ⟶ CC_{∗}^{per}(A)

or

U(𝔤𝔩(A[ϵ, η])) ⟶ End(CC_{∗}^{per}(A))  (5.2)

where [ϵ, η] = [ϵ, ϵ] = [η, η] = 0, deg ϵ = deg η = −1 and the differential in the L.H.S. is \frac{∂}{∂ϵ} + \frac{∂}{∂η}. For modules over any Lie algebra 𝔤 we shall write ⊗_{𝔤} instead of ⊗_{U(𝔤)}. \underline{Lemma 5.1}. For any Lie Algebra 𝔤 and any right (left) module M the map (5.2) descends to a morphism

k ⊗_{𝔤𝔩(k[η])} U(𝔤𝔩(A[ϵ, η]))_{𝔤𝔩(k[ϵ])} ⊗ k ⟶ \underline{End} (CC_{∗}^{per}(A))

\underline{Proof} First it is easy to see that the map descends to k ⊗_{𝔤𝔩(k[η])} U. In fact, this subalgebra acts by the operators L_{m} and L_{ad(m)} where m ∈ 𝔤𝔩(k), both are zero when followed by the trace map tr. To see that one can pass to k ⊗_{𝔤𝔩(k[η])} U(𝔤𝔩(A[ϵ, η]))_{𝔤𝔩(k[ϵ])} ⊗k note that L_{ad(m)} is zero on the image of i and that, because of the explicit formula in Definition 4.2, the operator J(ϵD_{1} · . . . · ϵD_{m}) is zero on the image of i if at least one of the D_{i} is of the form ad(m), m ∈ 𝔤𝔩(k). Indeed, this formula involves terms D_{i}(a_{j}) for any D_{i} which is a one-cochain. ■

Finally there is an explicit map

C_{∗}(𝔤𝔩(A[η]), 𝔤𝔩(k); k) ⟶ k ⊗_{𝔤𝔩(k[η])} U(𝔤𝔩(A[ϵ, η]))_{𝔤𝔩(k[ϵ])} ⊗ k  (5.3)

Let us construct this map. Let 𝔤 = 𝔤𝔩(A) and 𝔥 = 𝔤𝔩(k). To describe the image of a chain D_{1} ∧ ⋯ ∧ D_{n} ∧ ηE_{1} ∧ ⋯ ∧ ηE_{m}, write the expression

D_{1}(ϵ − η) · . . . · D_{n}(ϵ − η) · E_{1}ϵη · . . . · E_{m}ϵη

and then represent it as a sum

∑ ±D_{i_{1}}η · . . . · D_{i_{k}}η · E_{1}ϵη · . . . · E_{n}ϵη · D_{j_{1}}ϵ · . . . · D_{j_{m}}ϵ

in the symmetric algebra of the graded space 𝔤[ϵ, η]. For example D ↦ Dϵ − Dη;

D_{1} ∧ D_{2} ⟼ D_{1}η · D_{2}η + D_{1}η · D_{2}ϵ + (−1)^{(|D_{1}|+1)(|D_{2}|+1)}D_{2}η · D_{1}ϵ + D_{1}ϵ · D_{2}ϵ D_{1} ∧ D_{2}η ⟼ −D_{1}η · D_{2}ϵη + (−1)^{(|D_{1}|+1)(|D_{2}|+1)}D_{2}ϵη · D_{1}η

etc.

It turns out that this yields the morphism of complexes

C_{∗}(𝔤[η], 𝔥; k) ⟶ k ⊗_{𝔤𝔩(k[η])} U(𝔤𝔩(A[ϵ, η]))_{𝔤𝔩(k[ϵ])} ⊗ k  (5.4)

or, which is the same,

k ⊗_{𝔤𝔩(A[η])} U(𝔤𝔩(A[ϵ, η])_{𝔤𝔩(k[ϵ])} ⊗ k ⟶ k ⊗_{𝔤𝔩(k[η])} U(𝔤𝔩(A[ϵ, η]))_{𝔤𝔩(k[ϵ])} ⊗ k

\underline{Remark} . These maps are quasi-isomorphisms; one can easily show that the cohomology of both complexes is

C_{∗}(𝔤[η], 𝔥; k) = S(𝔥)_{𝔥} = H_{∗}(BH, k)

where H is the Lie group of the Lie algebra 𝔥 and BH is the classifying space of H. At the level of cohomology the inclusion

C_{∗}(𝔤, 𝔥; k) ⟶ C_{∗}(𝔤[η], 𝔥; k)  (5.5)

is the dual map to the Chern-Weil map

C^{∗}(𝔤, 𝔥; k) ⟵ W^{∗}(𝔥)_{basic};

W^{∗}(𝔥) = C_{∗}(𝔥[η], k) is the Weil algebra of 𝔥. Another realization of the map which is induced by 5.5) on homology is

k ⊗_{𝔥[ϵ]} U(𝔤[η]) ⊗_{U(𝔤)} U(𝔤[ϵ])_{𝔥[ϵ]} ⊗ k ⟶ k ⊗_{𝔥[η]} U(𝔤[ϵ, η])_{𝔥[ϵ]} ⊗ k  (5.6)

Section 6.

Index theorems

To illustrate our methods, we will sketch the proof of the algebraic index theorem from [Fe], [NT]. This proof, unlike the original ones, is designed to work in a much more general situation (families, foliations, manifolds with boundaries or with corners, complex varieties, etc.).

Let (M, ω) be a 2n-dimensional symplectic manifold with a star product ∗.

f ∗ g = ∑(iℏ)^{k}φ_{k}(f, g).

Let 𝔸^{ℏ}(M) = C^{∞}(M)[[ℏ]] with the product ∗. Put also 𝔸_{c}^{ℏ}(M) = C_{c}^{∞}(M)[[ℏ]]. The following is proved in [Fe], [NT]. Consider the canonical trace

Tr : 𝔸_{c}^{ℏ}(M) → ℂ[ℏ^{−1}, ℏ]]

constructed in [Fe], [NT]. One has

Tr(f) = \frac{1}{(iℏ)^{n}n!}[∫_{M} fω^{n} + ∑_{k=1}^{∞} (iℏ)^{k}D_{k}(f)ω^{n}]

where D_{k} are certain differential operators. All traces on 𝔸_{c}^{ℏ}(M) are proportional.

Let P^{2} = P, Q^{2} = Q be two elements of the matrix algebra M_{N}(𝔸^{ℏ}(M)) such that P − Q is in M_{N}(𝔸_{c}^{ℏ}(M)). Let P_{0} = P modℏ, Q_{0} = Qmodℏ, ch(P_{0}) = ∑_{k} \frac{1}{k!}trP_{0}(dP_{0})^{2k} (the Chern character of the connection dP_{0}d in the vector bundle P_{0}ℂ^{N}). We regard T M as a U(n)-bundle (since U(n) is a maximal compact subgroup of Sp(2n)).

Recall that for each deformation quantization of M a characteristic class θ is defined which is an element of H^{2}(M)[[ℏ]]: the class θ/iℏ is the curvature of a Fedosov connection defining the deformation (cf. [Fe], [NT]). \underline{Theorem 6.1}.

Tr(P − Q) = ∫_{M} (ch(P_{0}) − ch(Q_{0}))Â(T M)e^{θ/iℏ}

where θ is the characteristic class of ∗. \underline{Sketch of the proof}. Consider the characteristic map

χ : \overline{C}_{∗−1}^{λ}(𝔸^{ℏ}(M))[ℏ^{−1}] → \underline{End} CC_{∗}^{per}(𝔸_{c}^{ℏ}(M))[ℏ^{−1}] Note that the right hand side is the space of global sections of a fine sheaf ( the left hand side is not). Therefore one can extend χ to the morphism to the Chech - cyclic double complex:

χ : Č^{−∗}(\overline{C}_{∗−1}^{λ}(𝔸^{ℏ}(M)))[ℏ^{−1}] → \underline{End} CC_{∗}^{per}(𝔸_{c}^{ℏ}(M))[ℏ^{−1}]  (6.1)

Next step is to construct the fundamental class Ω in the left hand side of (6.1). Note that for any contractible Darboux chart U_{0}

\overline{HC}_{2n−1}(𝔸^{ℏ}(U_{0}))[ℏ^{−1}] ˜→ ℂ[ℏ^{−1}, ℏ]]  (6.2)

(the canonical generator is represented by the cycle

Ω_{0} = \frac{1}{2n(iℏ)^{n}} Alt(ξ_{1} ⊗ x_{1} … ξ_{n} ⊗ x_{n})  (6.2.1)

where alternation is over the group Σ_{2n} and [ξ_{i}, x_{j}] = iℏδ_{i}^{j});

\overline{HC}_{i}(𝔸^{ℏ}(U_{0}))[ℏ^{−1}] = 0  (6.3)

for i > 2n − 1.

Using the spectral sequence E_{ij}^{2} = H^{−i}(M, \overline{HC}_{j}(𝔸^{ℏ})), we construct a cycle of the complex Č^{−∗}(\overline{C}_{∗−1}^{λ}(𝔸^{ℏ}(M))[ℏ^{−1}] whose homology class, being restricted to any contractible Darboux chart, gives the canonical generator of \overline{HC}_{2n−1}. Let α be a periodic cyclic chain of 𝔸_{c}^{ℏ}(M); α_{0} = αmodℏ is a periodic cyclic chain of C_{c}^{∞}(M). (For non-unital algebras the periodic cyclic complex is defined as the kernel Ker(CC^{per}(Ã) → CC^{per}(k)) were Ã is the algebra A with adjoined unit). One proves that

Tr(χ(Ω))(α) = ∫_{M} μ(α_{0})modℏ  (6.4)

where μ : CC_{∗}^{per}(C_{c}^{∞}(M)) → (Ω^{∗}(M), d) is the morphism of complexes given by

μ(a_{0} ⊗ … ⊗ a_{k}) = \frac{(−1)^{k}}{k!} a_{0}da_{1} … da_{k}

([L]). Formula (6.4) follows easily from (6.2.1) and from the fact that [f, g] = iℏ{f, g}modℏ^{2}.

Now consider the cycles (k − 1)!η^{⊗k} in Č^{−∗}(\overline{C}_{∗−1}^{λ}(𝔸^{ℏ}(M)[η]))[ℏ^{−1}]. One has

χ((k − 1)!η^{⊗k}) = S^{k}  (6.5)

where S is the Bott isomorphism. We know from (6.4) the composition of the canonical trace with χ(Ω). To compute the canonical trace itself we have to compare the map χ(Ω) to the identity map. In view of (6.5), it is enough to express Ω in terms of the classes (k − 1)!η^{⊗k}. \underline{Theorem 6.2}. In Č^{−∗}(\overline{C}_{∗−1}^{λ}(𝔸^{ℏ}(M)[η])[ℏ^{−1}] one has

Ω ≃ ∑_{k}(−1)^{k}(Â(T M)e^{θ/iℏ})_{2k}^{−1} · (n + k − 1)!η^{⊗(n+k)}

Note that, since \underline{End} (CC_{∗}^{per}(𝔸_{c}^{ℏ}(M))) is the space of global sections of a fine sheaf, its cohomology is a module over the algebra H^{∗}(M, ℂ) : if (c_{U_{0}U_{1}...U_{p}}) is in Č^{p}(M, ℂ) then \underline{End} (CC_{∗}^{per}(𝔸_{c}^{ℏ}(M)))

c · α = ∑ c_{U_{0}U_{1}...U_{p}}ρ_{U_{0}}[B + b, ρ_{U_{1}}] … [B + b, ρ_{U_{p}}]

where {ρ_{U}} is a partition of unity.

We see that for any periodic cyclic cycle α of CC_{∗}^{per}(𝔸_{c}^{ℏ}(M))

Tr(e^{−θ/iℏ} · α) = ∫_{M} μ(α_{0})Â(T M)mod(ℏ)  (6.6)

Indeed,

Tr(e^{−θ/iℏ} · α) = Tr ∑_{k}(−1)^{k}(Â(T M)e^{θ/iℏ})^{2k} · S^{n+k}χ(Ω)(e^{−θ/iℏ} · α)

by Theorem 6.2; now we use (6.4).

We apply (6.6) to α = ch(P) − ch(Q) where ch is the Connes-Karoubi Chern character cycle in the periodic cyclic complex:

ch(P) = tr(P + ∑_{k≥1}(−1)^{k} \frac{(2k)!}{k!}(P − \frac{1}{2}) ⊗ P^{⊗2k})  (6.7)

([L]). Note that μ(ch(P)) = ch(P_{0}). The left hand side of the last formula stands for the Chern character of the vector bundle ImP_{0}.

It remains to show that for a periodic cyclic cycle α of 𝔸^{ℏ}(M) over ℂ (i.e when all the tensor products are taken over ℂ, not ℂ[[ℏ]]) one has

\frac{∂}{∂ℏ}Tr(e^{−θ/iℏ} · α) = 0  (6.8)

This can be done by passing to a one-parameter family of deformations

f ∗_{λ} g = ∑(iℏλ)^{k}φ_{k}(f, g).

One gets the deformed algebra 𝔸^{ℏ}(M × ℝ^{1}).

It turns out that for any deformed family M → B of symplectic manifolds one can construct the canonical morphism of complexes ([NT])

Tr_{θ} : CC_{∗}^{per}(𝔸_{c}^{ℏ}(M)) → Ω_{c}^{∗}(B)[ℏ^{−1}, ℏ]]

such that the Ω^{0} component is, for b ∈ B,

Tr_{θ}(α)(b) = Tr(α(b) · e^{−θ_{b}/iℏ})

where α(b) is the restriction of α to 𝔸_{c}^{ℏ}(M_{b}) and θ_{b} is the characteristic class of 𝔸^{ℏ}(M_{b}).

It remains to say a few words about the proof of Theorem 6.2. To prove it one has to construct and to compare various cocycles (cochains) of the complex Č^{∗}(M, L^{∗}) where L^{∗} is the sheaf of complexes \overline{C}_{−∗}^{λ}(𝔸^{ℏ}(M))[ℏ^{−1}]. If L^{∗} were a constant sheaf then one would be able to construct such cochains, for example the ones representing characteristic classes of the tangent bundle. One observes that L^{∗} is constant up to homotopy in the following sense. If 𝔤 is the Lie algebra of infinitesimal coordinate changes (in our case 𝔤 = Der(𝔸^{ℏ})) then for any open contractible chart U the action of 𝔤(U) on L^{∗}(U) is homotopically trivial; moreover, for X from 𝔤, let L_{X} : L^{∗} → L^{∗} be the action of 𝔤 on L^{∗}; one can also construct operators i_{X} : L^{∗} → L^{∗−1} satisfying the usual formulas:

[L_{X}, i_{Y}] = i_{[X,Y]}; [i_{X}, i_{Y}] = 0; [d, i_{X}] = L_{X}

In our case X = \frac{1}{iℏ}ad(a) and i_{X} = L\underline{a}.

Let 𝔸̂^{ℏ}(ℝ^{2n}) be the Weyl algebra ℂ[[ξ_{1}, …, ξ_{n}, x_{1}, …, x_{n}, ℏ]] with the Moyal product. Let 𝔤̂ = Der(𝔸̂^{ℏ}(ℝ^{2n})); then 𝔲(n) is a Lie subalgebra of 𝔤̂. Put

𝕃^{∗} = \overline{C}_{−∗}^{λ}(𝔸̂^{ℏ}(ℝ^{2n}))[ℏ^{−1}]

Let W^{∗}(𝔲(n)) be the Weil algebra. As a partial case of a general construction one gets a cochain map

[W^{∗}(𝔲(n)) ⊗ 𝕃^{∗}]_{basic} → Č(M, L^{∗})  (6.9)

The subscript basic stands for {α : L_{X}α = i_{X}α = 0, X ∈ 𝔲(n)}.

To finish the proof of the theorem one constructs the fundamental class Ω in [W^{∗}(𝔲(n)) ⊗ 𝕃^{∗}]_{basic} and proves an analogue of Theorem 6.2 by an explicit calculation. ■

Let us finish by saying a few words about how to carry out all the above computations explicitly at the level of cochains. Recall that one can realize any deformation 𝔸^{ℏ}(M) as the space of horizontal sections of a Fedosov connection D : Ω^{0}(M, 𝕎) → Ω^{1}(M, 𝕎) where 𝕎 is the Weyl bundle ([Fe]). Then, instead of the Cech complex Č^{∗}(M, L^{∗}), one considers the de Rham complex Ω^{∗}(M, 𝕃^{∗}); all the cochains which participate in the proof have an easy explicit Chern - Weil style realization in this complex.

\underline{Section 7.} \underline{Examples}

For any differential graded algebra B there is a natural gradind on the Hochschild chain complex C_{∗}(B, B). We denote by C_{∗}(B, B)^{j} the space of homogeneous elements of degree j in C_{∗}(B, B).Let

\begin{matrix}C^{n}(A) = ⨁_{i+j=n} C_{−i}(E_{A}^{∗}, E_{A}^{∗})^{j} \\ C_{∞}^{n}(A) = ∏_{i+j=n} C_{−i}(E_{A}^{∗}, E_{A}^{∗})^{j} \end{matrix}

We have defined a homotopically associative • product on these complexes; their homology is denoted by H^{∗}(A), resp. by H_{∞}^{∗}(A).

\underline{Theorem 7.1}. Let A = k[X] be the algebra of regular functions on an affine algebraic variety X over a field k of characteristic zero. Then

H^{∗}(A) ˜→ D(Ω_{X/k}^{∗})

(the ring of all differential operators on Ω);

H_{∞}^{∗}(A) ˜→ End(Ω_{X/k}^{∗})

Note that HH_{∗}(A) ≃ Ω_{X/k}^{∗}; the • action of H^{∗} on HH^{∗} is the obvious one.

\underline{Proof of 7.1}.

\underline{Lemma 7.2}. Let E^{∗}, F^{∗} be two differential graded algebras together with a quasiisomorphism f : E^{∗} → F^{∗}. Then f induces an quasi-isomorphism

⨁_{i+j=n} C_{−i}(E^{∗}, E^{∗})^{j} ˜→ ⨁_{i+j=n} C_{−i}(F^{∗}, F^{∗})^{j}

\underline{Proof}. The map induced by f is an isomorphism at the level of E^{1} terms of the spectral sequences with E^{1} = ⨁_{i+j=n} C_{−i}(H^{∗}(E^{∗}), H^{∗}(E^{∗}))^{j}, resp.

⨁_{i+j=n} C_{−i}(H^{∗}(F^{∗}), H^{∗}(F^{∗}))^{j}, the Hochschild complex of the graded algebra of cohomology of E^{∗}, resp. F^{∗}. Note that we are able to apply this argument to those converging spectral sequences (this is not true if one replaces ⨁ by ∏). ■

Note that by [HRK] E_{A}^{∗} is quasi-isomorphic to F^{∗} = Γ(X, ⋀^{∗} T X) as differential graded algebras. Locally, F_{U}^{∗} = O_{U} ⊗ ⋀^{∗}(∂_{x_{1}}, …, ∂_{x_{n}}); HH_{∗}(F_{U}^{∗}) = Ω_{U/k}^{∗} ⊗ ⋀^{∗}(∂_{x_{1}}, …, ∂_{x_{n}})⊗S^{∗}(d∂_{x_{1}}, …, d∂_{x_{n}}) (classes d∂_{x_{i}} are represented by cycles (1, d∂_{x_{i}})). Now, looking at the • action on HH^{∗}(O_{U}) ≃ Ω_{U/k}^{∗}, one sees that Ω_{U/k}^{∗} acts by multiplication, ∂_{x_{i}} acts by contraction i_{∂_{x_{i}}} and d∂_{x_{i}} acts by Lie derivative L_{∂_{x_{i}}} . Whence the local isomorphism

H^{∗}(O_{U}) ˜→ D(Ω_{U/k}^{∗});

this statement may be globalized exactly as in [HRK] Now let us compute H_{∞}^{∗}(A). We have to use the other spectral sequence whose E_{1} term is HH_{∗}(E_{A}^{∗}) , the Hochschild homology of the graded anlebra E_{A}^{∗} (if one forgets about the differential δ).

For the vector spaces V and W we will denote by T (V′)ˆ⊗W the space of multilinear maps from V to W. We will use similar notation for certain subfactors of T (V′).

As a graded algebra, E_{A}^{∗} is just T^{∗}(A′)ˆ⊗A by which we mean the space of all multilinear maps from A to A. One can show that

E_{2}^{−p,n} = HC^{n−1}(A)ˆ⊗Ω^{p} + HC^{n−2}(A)ˆ⊗Ω^{p+1}
E_{3}^{−p,n} = (KerS)^{n−1} ˆ⊗Ω^{p} + (CokerS)^{n−2} ˆ⊗Ω^{p+1}

where S : HC^{n} → HC^{n+2} is the Bott operator ([L]). Recall the Gysin exact sequence ([C], [T], [L]):

0 ⟶ (CokerS)^{n} ⟶ HH^{n}(A) ⟶ (kerS)^{n−1} ⟶ 0

It is not hard to construct for any element of HH^{∗}(A)ˆ⊗Ω_{X/k}^{∗} a corresponding cycle of the complex C^{∗}. This shows that the spectral sequence degenerates at E_{3} term and that H_{∞}^{∗}(A) ˜→ HH^{∗}(A)ˆ⊗Ω_{X/k}^{∗}, ■

Section 8.

Characteristic cochain of a flat element.

Let ∇ be an element of C^{∗}(A, A) such that |∇| is odd (and therefore deg ∇ is even). Assume that

δ∇ + \frac{1}{2}[∇, ∇] = 0  (8.1)

Consider an element

e^{∇} = ∑ \frac{1}{n!} ∧^{n} ∇ ∈ ∧^{∗}𝔤(A)

in the completed space ∏ R_{k}(𝔤(A)). Put

X(∇) = J(e^{∇} ⊗ 1)  (8.2)

This is a well-defined operator

X(∇) : CC_{∗}^{per,(0)}(A) → CC_{∗}^{per}(A)

where CC_{∗}^{per,(0)} is the subcomplex of finite cochains:

CC_{n}^{per,(0)}(A) = ⨁_{i≡n(2)}A ⊗ \overline{A}^{⊗n},

while, as usually,

CC_{n}^{per}(A) = ∏_{i≡n(2)}A ⊗ \overline{A}^{⊗n}.

\underline{Theorem 8.1}.

[B + b, X(∇)] = X(∇) · L_{∇}.

\underline{Proof}. Indeed, in the completion of ∧^{∗}𝔤(A) ⊗ U(𝔤(A)):

\begin{matrix}(δ + ∂^{Lie})(e^{∇} ⊗ 1) = e^{∇}(δ∇ + \frac{1}{2}[∇, ∇])⊗ \\ ⊗1 + e^{∇} ⊗ L_{∇}. ■ \end{matrix}

In some cases there is a natural topology on A such that X(∇) converges and defines an operator on a complex larger than CC^{per,0}.

The formula for X(∇) is obtained as follows. In the periodic cyclic complex of the algebra generated by the even elements ∇ and e^{−t∇}, t ≥ 0, put

e^{I\underline{∇}} = ∑_{n≥0} ∫_{\begin{matrix}t_{0}+···+t_{n}=1 \\ t_{i}≥0 \end{matrix}}e^{t_{o}i_{∇}}S\underline{∇}e^{t_{1}i_{∇}} … S\underline{∇}e^{t_{n}i_{∇}}dt … dt_{n−1}

e^{ti\underline{∇}}(D_{0}, …, D_{m}) = (D_{0} ⌣ e^{t∇}, D_{1}, …, D_{m});

S\underline{∇}(D_{0}, …, D_{m}) = n ; ∑_{j=0} j−1 ; ∑_{i=0}(−1)^{jn+n+j+i}(1, D_{j}, …, D_{m},

D_{0}, …, D_{i}, ∇, …, D_{j−1})

Then

X(∇)(α) = α • (e^{I\underline{∇}}1)

where • is the product from Theorem 3.1. One has e^{I\underline{∇}}1 = e^{∇} − ∫ (e^{t_{0}∇}, e^{t_{1}∇}, ∇)+

+ ∫ [(2(e^{t_{0}∇}, e^{t_{1}∇}, ∇, e^{t_{2}∇}, ∇)−
−(e^{t_{0}∇}, ∇, e^{t_{1}∇}, ∇, e^{t_{2}∇}) + (e^{t_{0}∇}, ∇, e^{t_{1}∇}, e^{t_{2}∇}, ∇)] + …

Section 9.

Bivariant JLO cochain.

In [JLO] and [GS] a Chern character of a θ-summable Fredholm module over a Banach algebra A was defined. It is given by a cocycle of the complex dual to CC^{per,0}(A); this cocycle satisfies a special growth condition which makes it an entire cyclic cocycle in the sense of Connes. Here we will construct its bivariant version, which means, a homomorphism from CC^{per,0}(A) to CC^{per}(A) whose composition with a trace is given by the formula from [JLO] and [GS].

Given a ℤ_{2}-graded algebra A and an odd element D / of A, put

∇ = ad(D /) − D /^{2};  (9.1)

then ∇ satisfies (8.1).

\underline{Theorem 9.1}. Put

ch(D /) = X(∇) · e^{L_{D̸}};

then

[B + b, ch(D /)] = 0  (9.2)

\underline{Proof}. We have seen that

[B + b, X(∇)] = X(∇)L_{∇};  (9.3)

it suffices to show that

[B + b, e^{L\underline{D̸}}] = −L_{∇} · e^{L\underline{D̸}}.  (9.4)

Indeed,

[B + b, e^{L\underline{D̸}}] = − ∫_{0}^{1} e^{tL\underline{D̸}}L_{adD̸}e^{(1−t)L_{D̸}}dt =
= −L_{adD̸} · e^{L_{D̸}} + ∫ t L_{[\underline{D̸D̸}]}e^{L_{D̸}}dt =
= −L_{∇}e^{L\underline{D̸}}, ■

Section 10.

“The Fukaya category”.

In this Section we extend the results of Section 2 by “moving away from the diagonal”. For any automorphism α of an algebra A we define the twisted Hochschild complex C_{∗}(A, A_{α}) (related to noncommutative geometry of fixed points of α) and construct in these terms a homotopically associative category for which C^{∗}(A) or C_{∞}^{∗}(A) is the ring of endomorphisms of the identity.

Let α, β be two automorphisms of A. Define the new bimodule _{α}A_{β} over A as follows: _{α}A_{β} = A as k-modules and a · m · b = α(a)mβ(b) for a, m, b in A. Consider the chain complex C_{∗}(A, _{α}A_{β}) as in Section 2 and the cochain complex αE_{β}^{∗} = C^{∗}(A, _{α}A_{β}) as in Section 1. We shall write A_{α} =_{id} A_{α}. The cup product on Hochschild cochains is well defined as a morphism _{α}E_{β}^{∗} ⊗_{β} E_{γ}^{∗} →_{α} E_{γ}^{∗}.

Given an automorphism α of A, one can define its action on a Hochschild cochain from E_{A}^{∗} in two ways:

(αD)(a_{1}, …, a_{n}) = α(D(a_{1}, …, a_{n}));
(Dα)(a_{1}, …, a_{n}) = D(αa_{1}, …, αa_{n})

Both yield morphisms of complexes E_{A}^{∗} →_{α} E_{α}. We define an E_{A}^{∗}-bimodule structure on _{α}E_{β}^{∗} as follows:

D · M · E = Dα ⌣ M ⌣ βE

for D, E ∈ E_{A}^{∗} and M ∈_{α}E_{β}^{∗}.

\underline{Theorem 10.1}. The pairings (0.6) can be extended to the homotopically associative natural morphisms of complexes

• : C_{∗}(A, A_{α}) ⊗ C_{∗}(E_{A}^{∗},_{α} E_{β}^{∗}) → C_{∗}(A, A_{β})  (10.1)
• : C_{∗}(E_{A}^{∗}, _{α}E_{β}^{∗}) ⊗ C_{∗}(E_{A}^{∗}, _{β}E_{γ}^{∗}) → C_{∗}(E_{A}^{∗}, _{α}E_{γ}^{∗})  (10.2)

\underline{Proof} As in Section 2, put for a ∈ C_{∗}(A, A_{α}) and x ∈ C_{∗}(E_{A}^{∗}, _{α}E_{β})

a • x = a •_{1} x + a •_{2} x

where (a_{0}, …, a_{n}) •_{1} (D_{0}, …, D_{m}) = ∑ ±(a_{0} · D_{0}(a_{1}, …), …, a_{i_{1}}, D_{1}(a_{i_{1}+1}, …), …, D_{m}(a_{i_{m}+1}, …), …) (10.3)

(a_{0}, …, a_{n}) •_{2} (D_{0}, …, D_{m}) =

= ∑_{q≤n+1} ±(D_{m}(a_{q}, …, a_{n}, a_{0}, αa_{1}, …, αa_{i_{0}}) · D_{0}(a_{i_{0}+1}, …), …, A_{i_{1}},

, D_{1}(a_{i_{1}+1}, …), …, D_{m−1}(a_{i_{m−1}+1}, …), …) (10.4)

For A ∈ C_{∗}(E_{A}^{∗}, _{α}E_{β}) and x ∈ C_{∗}(E_{A}^{∗}, _{β}E_{γ})

A • x = A •_{1} x + A •_{2} x  (10.5)

where (A_{0}, …, A_{n}) •_{1} (D_{0}, …, D_{m}) = ∑ ±(A_{0} ⌣ D_{0}{A_{1}, A_{2}, … }, …, A_{i_{1}}, D_{1}{A_{i_{1}+1}, … }, …, D_{m}{A_{i_{m}+1}, … }, …) (10.6)

(A_{0}, …, A_{n}) •_{2} (D_{0}, …, D_{m}) =

= ∑_{q≤n+1} ±(D_{m}{{A_{q}, …, A_{n}, A_{0}, A_{1}, …, A_{i_{0}}}} ⌣ D_{0}{A_{i_{0}+1}, … }, …, A_{i_{1}},

, D_{1}{A_{i_{1}+1}, … }, …, D_{m−1}{A_{i_{m−1}+1}, … }, …) (10.7)

and

D_{m}{{A_{q}, …, A_{n}, A_{0}, A_{1}, …, A_{i_{0}}}}(a_{1}, …, a_{n}) =

= ∑ ±D_{m}(αa_{1}, …, A_{q}(αa_{j}, …,), …,

, αa_{t}, A_{0}(a_{t+1}, …,), βa_{p}, …, βA_{1}(a_{r}, …), βa_{s}, …)

One checks that these maps are homotopically associative morphisms of complexes. ■

One gets the cohomology groups H^{∗}(α, β) and H_{∞}^{∗}(α, β) which form a category. When A is the ring of functions on X then, in simple cases, the groups H^{∗}(α, β) are results of some standard D-module constructions on T^{∗}(X). When A is a de-

formed algebra of functions on a symplectic manifold M then , when M is simply

connected, all the automorphisms of A are of the form exp(ad\frac{1}{iℏ}f); H^{∗}(α, β) and

H_{∞}^{∗}(α, β) reflect the geometry of fixed point sets of α and β and of related loop

and path spaces. It looks as if there was, along with H^{∗} and H_{∞}^{∗}, the intermediate

semi-infinite cohomology which is more closely related to Floer cohomology of loop

spaces.

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