Gerstenhaber algebra

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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.

[edit] 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.

[edit] 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

[edit] References

Gerstenhaber, Murray The cohomology structure of an associative ring. Ann. of Math. (2), vol. 78 (1963), 267-288. MR 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.