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# Gerstenhaber algebra

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.

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) (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.

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 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. http://jstor.org/stable/1970343.
* Getzler, E. (1994). "Batalin-Vilkovisky algebras and two-dimensional topological field theories". Communications in Mathematical Physics 159 (2): 265–285. doi:10.1007/BF02102639.
* Kosmann-Schwarzbach, Y. (2001), "Poisson algebra", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

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