Gerstenhaber algebra

In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder-Weyl theory as the algebra of generalized Poisson brackets defined on differential forms.

Definition

A Gerstenhaber algebra is a 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

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

Examples

References

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