## Gerstenhaber algebras and BV-algebras in Poisson geometry.(English)Zbl 0941.17016

A Gerstenhaber algebra consists of a triple $$({\mathcal A}=\bigoplus_{i\in{\mathbb Z}}{\mathcal A}^i,\wedge, [\cdot,\cdot])$$ such that $$({\mathcal A},\wedge)$$ is a graded commutative associative algebra, and $$({\mathcal A} =\bigoplus_{i\in{\mathbb Z}}{\mathcal A}^{(i)},[\cdot,\cdot])$$, with $${\mathcal A}^{(i)}={\mathcal A}^{i+1}$$, is a graded Lie algebra, and $$[a,\cdot]$$, for each $$a\in{\mathcal A}^{(i)}$$ is a derivation with respect to $$\wedge$$ of degree $$i$$.
An operator $$D$$ of degree $$-1$$ is said to generate the Gerstenhaber algebra bracket if for every $$a\in{\mathcal A}^k$$ and $$b\in{\mathcal A}$$, $[a,b]=(-1)^k(D(a\wedge b)-Da\wedge b-(-1)^ka\wedge Db).$ If the generating operator $$D$$ can be found of square zero, the Gerstenhaber algebra is called exact or a Batalin-Vilkovisky algebra (BV-algebra in short). The well-known examples are: the algebra of multivector fields with the Schouten bracket and the algebra of differential forms on a Poisson manifold with the Koszul bracket.
Let $$A$$ be a vector bundle of rank $$n$$ over the base $$M$$, and let $${\mathcal A}=\bigoplus_{0\leq k\leq n}\Gamma(\bigwedge^kA)$$ be its corresponding exterior algebra. It is known that putting a Gerstenhaber bracket on $$\mathcal A$$ is equivalent to putting a Lie algebroid structure on $$A$$.
The main result of the paper states that there is a one-one correspondence between $$A$$-connections on the line bundle $$\bigwedge^nA$$ and linear operators $$D$$ generating the Gerstenhaber bracket on $$\mathcal A$$. Under this correspondence, flat connections correspond to operators of square zero. Introducing the homology associated with flat connections ($$D^2=0$$), the author proves the duality between Poisson homology and cohomology: on an orientable unimodular Poisson manifold $$(P,\pi)$$, $H_*(P,\pi)\cong H^{n-*}_\pi(P).$

### MSC:

 17B63 Poisson algebras 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17B62 Lie bialgebras; Lie coalgebras 53D17 Poisson manifolds; Poisson groupoids and algebroids

### Keywords:

BV-algebra; Poisson manifold; modular form; cohomology
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