A Gerstenhaber algebra is a graded-commutative algebra with a Lie bracket of degree −1 satisfying the Poisson identity. Everything is understood to satisfy the usual superalgebra sign conventions. More precisely, the algebra has two products, one written as ordinary multiplication and one written as [,], and a Z-grading called degree (in theoretical physics sometimes called ghost number). The degree of an element a is denoted by |a|. These satisfy the identities
(ab)c = a(bc) (The product is associative)
ab = (−1)|a||b|ba (The product is (super) commutative)
|ab| = |a| + |b| (The product has degree 0)
|[a,b]| = |a| + |b| − 1 (The Lie bracket has degree −1)
[a,b] = −(−1)(|a|−1)(|b|−1) [b,a] (Antisymmetry of Lie bracket)
[a,[b,c]] = [[a,b],c] + (−1)(|a|−1)(|b|−1)[b,[a,c]] (The Jacobi identity for the Lie bracket)
Gerstenhaber algebras differ from Poisson superalgebras in that the Lie bracket has degree −1 rather than degree 0. The Jacobi identity may also be expressed in a symmetrical form
Examples
Gerstenhaber showed that the Hochschild cohomology H*(A,A) of an algebra A is a Gerstenhaber algebra.
A Batalin–Vilkovisky algebra has an underlying Gerstenhaber algebra if one forgets its second order Δ operator.