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. Definition 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 Zgrading (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. Examples * 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.
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