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

Xu, Ping

Commun. Math. Phys. 200, No. 3, 545-560 (1999).

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 \((P,\pi)\), \[ H_*(P,\pi)\cong H^{n-*}_\pi(P). \]

Reviewer: Janusz Grabowski (Warszawa)

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