Gerstenhaber algebra

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 Z-grading (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"] | = |"a"| + |"b"| - 1 (The Lie bracket has degree -1)
*("ab")"c" = "a"("bc"), "ab" = (−1)^|"a"||"b"|"ba" (the product is associative and (super) commutative)
* ["a","bc"] = ["a","b"] "c" + (−1)^{|"a"|(|"b"|-1)}"b" ["a","c"] (Poisson identity)
* ["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"] (Jacobi identity for Lie bracket)

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.
*A Batalin-Vilkovisky algebra has an underlying Gerstenhaber algebra if one forgets its second order differential operator.
*The exterior algebra of a Lie algebra is a Gerstenhaber algebra.
*The differential forms on a Poisson manifold form a Gerstenhaber algebra.
*The multivector fields on a manifold form a Gerstenhaber algebra using the Schouten-Nijenhuis bracket

References

*Gerstenhaber, Murray [http://links.jstor.org/sici?sici=0003-486X%28196309%292%3A78%3A2%3C267%3ATCSOAA%3E2.0.CO%3B2-I "The cohomology structure of an associative ring."] Ann. of Math. (2), vol. 78 (1963), 267-288. MathSciNet|id=0161898
*E. Getzler "Batalin-Vilkovisky algebras and two-dimensional topological field theories" DOI|10.1007/BF02102639 Commun. Math. Phys., vol. 159, no. 2 (1994), 265-285.

*springer|id=p/p110170|title=Poisson algebra|author=Y. Kosmann-Schwarzbach