Cohomology and Deformation of Leibniz Pairs

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra A together with a Lie algebra L mapped into the derivations of A. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential. The importance of Poisson algebras in classical mechanics makes it useful to have a deformation theory for them. To construct this we are led to define a more general concept: A Leibniz pair (A, L) consists of an associative algebra A and a Lie algebra L over some common coefficient ring k, connected by a Lie algebra morphism µ : L → Der A, the Lie algebra of k-linear derivations of A into itself. It can happen that L is identical with A as a k-module; but note that in the graded case discussed below the gradings may differ. When this is so, denoting the associative product of a, b ∈ A by ab and their Lie product by [a, b], if one has further that µ(a)(bc) = [a, bc] = [a, b]c+b[a, c], then A will be called a non-commutative Poisson algebra (" ncPa "). A Poisson algebra in the usual sense is one where the associative multiplication on A is commutative (and k = R or C). A Leibniz pair with A = L need not be a Poisson algebra since one may have, for example, an abelian Lie multiplication together with a non-trivial structure morphism µ. This notion of a non-commutative Poisson algebra is a particular case of that suggested by P. Xu [21]. In his definition, he considers an associative algebra A together with a class Π ∈ H 2 (A, A) such that [Π, Π] = 0 in H 3 (A, A) in the sense of the Gerstenhaber bracket on Hochschild cohomology of A. Xu's definition is especially suitable for the geometric situation, such as A = C[x 1 ,. .. , x n ] or A = C ∞ (M), when M is a manifold and one takes multilinear differential operators as cochains. Then H 2 (A, A) will be naturally isomorphic to the space of bivector fields and the condition [Π, Π] = dB, where, due Hodge theory for Hochschild cohomology, one can assume B is symmetric; this will yield the Jacobi identity for Π. Thus, Π will give rise to a Poisson bracket on A. …