Gerstenhaber Algebra

A Gerstenhaber algebra is a differential 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| = |a| + |b| (The product has degree 0)
• || = |a| + |b| - 1 (The Lie bracket has degree -1)
• (ab)c = a(bc) (The product is associative)
• ab = (−1)|a||b|ba (The product is (super) commutative)
• = c + (−1)(|a|-1)|b|b (Poisson identity)
• = −(−1)(|a|-1)(|b|-1) (Antisymmetry of Lie bracket)
• ] = ,c] + (−1)(|a|-1)(|b|-1)] (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