In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism.
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 (sometimes called ghost number). The degree of an element a is denoted by |a|. These satisfy the identities
Gerstenhaber algebras differ from Poisson superalgebras in that the Lie bracket has degree -1 rather than degree 0.
* Gerstenhaber showed that the Hochschild cohomology H*(A,A) of a graded algebra A is a Gerstenhaber algebra.
* Gerstenhaber, Murray (1963). "The cohomology structure of an associative ring". Ann. of Math. 78 (2): 267–288. doi:10.2307/1970343. http://jstor.org/stable/1970343.