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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. 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.

Contents

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

  • |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

  • Gerstenhaber showed that the Hochschild cohomology H*(A,A) of an algebra A is a Gerstenhaber algebra.
  • A Batalin–Vilkovisky algebra has an underlying Gerstenhaber algebra if one forgets its second order Δ 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 algebra Wikipedia
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