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3.4.1 Hochschild cohomology

Let us review the theory of Hochschild cohomology, see [50] or [12] Section 2 for references.

dened as follows:

Cn(A, A) :=Hom_C(A⊗n, A), n_>0. (3.26)

The dierential dHis dened on homogeneous elements f ∈Cn_{(A, A)by the formula}

(dH(f))(a0, a1, . . . , an) :=a0f(a1, . . . , an) + n X k=1 (−1)kf(a0, . . . , ak−1ak, . . . , an) +(−1)n+1f(a0, . . . , an−1)an. (3.27)

We see that dHf ∈ Cn+1_{(A, A). We can prove dH◦dH = 0 therefore C}•_{(A, A) is a} cochain complex.

The Hochschild cohomology of A is dened as the cohomology group of the cochain

complex C•(A, A), and we denote it by HH•(A, A) or for short HH•(A):

HHn

(A) := Hn(C•(A, A)). (3.28)

Now let us look at the case n= 2. The following observation is easy to get:

Proposition 3.4.1. Letf ∈C2_{(A, A) =}Hom

C(A⊗A, A). Thenf is a 2-coboundary

if and only if there exists a g ∈C1(A, A) = Hom_C(A, A) such that for any a, b∈A

f(a, b) =ag(b)−g(ab) +g(g)b. (3.29)

Moreover, f is a 2-cocycle if and only if for any a, b,∈A

af(b, c)−f(ab, c) +f(a, bc)−f(a, b)c= 0. (3.30)

3.4.2 The Gerstenhaber bracket on Hochschild cochains and

cohomologies

In this section we give a quick review of the Gerstenhaber bracket. For more details and proofs see [17] or [8] Section 1. For further topics see the survey [15].

First, we dene an operation ◦ : Ck(A, A)⊗ Cl(A, A) → Ck+l−1(A, A). Let f1 ∈ Ck(A, A)and f2 ∈Cl(A, A),

(f1◦f2)(a1, . . . , ak+l−1) := = k−1 X i=0 (−1)(k−i−1)(l−1)f1(a1. . . , ai, f2(ai+1, . . . , ai+l), ai+l+1, . . . , ak+l−1). (3.31)

In particular, for 2-cochains we have

Proposition 3.4.2. Let f1, f2 ∈C2(A, A), then f1◦f2 ∈C3(A, A) and is given by

(f1◦f2)(a1, a2, a3) =f1(f2(a1, a2), a3)−f1(a1, f2(a2, a3)). (3.32)

In particular, for f ∈C2(A, A) we have

(f ◦f)(a1, a2, a3) =f(f(a1, a2), a3)−f(a1, f(a2, a3)). (3.33)

Proof: This is just the denition. 2

The Gerstenhaber bracket is dened to be

The Gerstenhaber bracket is a Lie bracket. In fact we have the following

Theorem 3.4.3. The operation "◦" gives a pre-Lie algebra structure onC•−1_{(A, A).}

Therefore we obtain that (C•−1_{(A, A),[, ]}

G) is a graded Lie algebra. Proof: See [17]. 2

Proposition 3.4.4. Let f ∈C2_{(A, A)}, then

[f, f]_G = 2f ◦f. (3.35)

Proof: We get this directly from the denitions. 2

The Gerstenhaber bracket is compatible with the Hochschild dierential dH. In fact dH is inner in the Gerstenhaber bracket. More precisely, let µ: A⊗A → A denote

the multiplication map inA. Thenµ∈C2_{(A, A). We have the following}

Proposition 3.4.5. For any f ∈Ck_{(A, A), we have}

dHf = [µ, f]_G∈Ck+1(A, A). (3.36)

We also have [µ, µ]_G = 0.

Proof: Compare the denition of dH in Equation 3.27 and the denition of the Ger- stenhaber bracket in Equation 3.31 and Equation 3.34. The fact that [µ, µ]_G = 0 is

exactly the associativity of µ. 2

As a result, we have the following theorem:

ential dH. In other words, for any f1 ∈Ck(A, A) and f2 ∈Cl(A, A), we have

dH([f1, f2]G) = [dHf1, f2]G+ (−1)k−1[f1,dHf2]G. (3.37)

Therefore the Gerstenhaber bracket reduces to the Hochschild cohomology HH•−1 (A).

Proof: Since dHis an inner derivation according to Propostion 3.4.5, Equation 3.37 is a consequence of the super-Jacobi identity of the graded Lie algebra(C•−1_{(A, A),[, ]}

G).

2

3.4.3 HH

_(A)

_{and the deformations of}

_A

The Hochschild cohomology plays an important role in the deformation theory, see [18] or [12] Section 2.

LetAbe an associative _Calgebra (infact we can replace _Cby any eld). A deforma-

tion of the algebra structure ofAmeans that we xAas aC-vector space and change

the multiplication operation onA. More precisely let_C[[t]]be the formal power series

of t and we dene

A[[t]] := A⊗_CC[[t]]. (3.38)

A[[t]] is obviously a _C[[t]]-module.

A deformation of the algebra structure on A is given by a map

m:A[[t]]⊗A[[t]]−→A[[t]] (3.39)

where m is required to be _C[[t]]-bilinear. So we only need to know the value of m on A⊗A.

For any a, b∈A, we can write m(a, b)as m(a, b) = ab+ ∞ X k=0 tkmk(a, b). (3.40)

We see that each mk belongs toC2(A, A).

Remark 29. The element t is called the deformation parameter. If we evaluate at t = 0 we get the original multiplication on A. On the other hand if we evaluate at t6= 0, omit the convergence problem, we get a new binary operationA⊗A →A.

Being a multiplication, m needs to satisfy the associativity law.

Theorem 3.4.7 (Formal deformation, see [18] Chapter I.1). Let m(a, b) = ab+

P∞

k=0t

k_m

k(a, b) as in Equation 3.40. Then m satises the associativity law if and

only if for each k_>1, we have

dHmk+ 1 2 k−1 X i=1 [mi, mk−i] = 0. (3.41)

If this holds, we say that m gives a formal deformation of A.

Proof: The associativity law means that for any a, b, c∈A, we have

m(a, m(b, c))−m(m(a, b), c) = 0. (3.42)

Now consider m as an element in C2_{(A[[t]], A[[t]])}, then Equation 3.42 is exactly

[m, m]_G = 0. (3.43)

We write m = µ+P∞

because we know [µ, f]_G = dHf and [µ, µ]_G = 0 in Proposition 3.4.5, Equation 3.43

becomes the Maurer-Cartan Equation

dH( ∞ X k=1 tkmk) + 1 2[ ∞ X k=1 tkmk, ∞ X k=1 tkmk]G= 0. (3.44)

In the expansion of Equation 3.44, we take the tk term and get Equation 3.41. 2

Corollary 3.4.8 (Innitesimal deformation). msatises the associativity law modt2

if and only if dHm1 = 0, i.e. for any a, b, c∈A, we have

am1(b, c)−m1ab, c+m1(a, bc)−m1(a, b)c= 0. (3.45)

If this holds, we say that m gives an innitesimal deformation of A.

Moreover, m satises the associativity law modt3 _{if and only if dHm}

1 = 0 together

with

dHm2+ 1

2[m1, m1]G= 0. (3.46)

The above equation is equivalent to

dHm2+m1◦m1 = 0 (3.47)

since in Proposition 3.4.4 we know that [m1, m1]G= 2m1◦m1

Proof: This is an direct corollary of Theorem 3.4.7. 2

trivial. In other words, wether or not we can nd an algebraic isomorphism

θ : (A[[t]], µ)−→(A[[t]], m) (3.48)

where θ is _C[[t]]-linear and is given by

θ(a) = a+

∞

X

k=1

tkθk(a). (3.49)

The requirement forθ is for any a, b∈A

θ(ab) = m(θ(a), θ(b)). (3.50)

The existence of θ is a complicated problem. First we have:

Proposition 3.4.9 (Innitesimally trivial deformation). There exists aθ1 ∈C1(A, A)

such that θ =id+tθ1 satisfyies Equation 3.50 modt2 if and only if m1 ∈B2(A, A).

If this holds, we say that m is an innitesimally trivial deformation of A.

Proof: We expand both sides of Equation 3.50 and look at the t term we get

θ1(ab) =θ1(a)b+aθ1(b) +m1(a, b) (3.51)

In other words

m1+dHθ1 = 0. 2 (3.52)

Further discussion of the triviality of deformations involves the concept of gauge equivalence of Maurer-Cartan elements, see [35] Section 1 or [37] Chapter 13.

3.5 The noncommutative Poisson bracket and the de-