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 Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism.

Definition[edit]

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

  • |ab| = |a| + |b| (The product has degree 0)
  • |[a,b]| = |a| + |b| - 1 (The Lie bracket has degree -1)
  • (ab)c = a(bc) (The product is associative)
  • ab = (−1)|a||b|ba (The product is (super) commutative)
  • [a,bc] = [a,b]c + (−1)(|a|-1)|b|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]] (The Jacobi identity for the Lie bracket)

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

(-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b|-1)}[c,[a,b]] = 0.\,

Examples[edit]

References[edit]