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

• |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, Murray (1963). "The cohomology structure of an associative ring". Ann. of Math. 78 (2): 267–288. doi:10.2307/1970343. JSTOR 1970343.
• Getzler, E. (1994). "Batalin-Vilkovisky algebras and two-dimensional topological field theories". Communications in Mathematical Physics. 159 (2): 265–285. arXiv:hep-th/9212043. Bibcode:1994CMaPh.159..265G. doi:10.1007/BF02102639.
• Kosmann-Schwarzbach, Y. (2001), "Poisson algebra", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Kanatchikov, Igor V. (1997). "On field theoretic generalizations of a Poisson algebra". Rep. Math. Phys. 40 (2): 225–234. arXiv:hep-th/9710069. doi:10.1016/S0034-4877(97)85919-8.