In mathematics and

theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an

algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...

discovered by Murray Gerstenhaber (1963) that combines the structures of a

supercommutative ring In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have :yx = (-1)^xy , where , ''x'', denotes the grade of the element and is 0 or 1 ( ...

and a

graded Lie superalgebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operati ...

. It is used in the

Batalin–Vilkovisky formalism In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, w ...

. It appears also in the generalization of Hamiltonian formalism known as the

De Donder–Weyl theory In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this frame ...

as the algebra of generalized

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

s defined on differential forms.

Definition

A Gerstenhaber algebra is a graded-commutative algebra with a

Lie bracket In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identit ...

of degree −1 satisfying the Poisson identity. Everything is understood to satisfy the usual

superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...

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'')''c'' = ''a''(''bc'') (The product is associative) *''ab'' = (−1)^{, ''a'', , ''b'',}''ba'' (The product is (super) commutative) *, ''ab'', = , ''a'', + , ''b'', (The product has degree 0) *, 'a'',''b'' = , ''a'', + , ''b'', − 1 (The Lie bracket has degree −1) * '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''">'b'',''c''.html" ;"title="'a'',[''b'',''c''">'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 In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...

may also be expressed in a symmetrical form :

+(-1)^[b,[c,a">,c.html" ;"title=",[b,c">,[b,c+(-1)^[b,[c,a+(-1)^[c,[a,b">,c">,[b,c<_a>+(-1)^[b,[c,a.html" ;"title=",c.html" ;"title=",[b,c">,[b,c+(-1)^[b,[c,a">,c.html" ;"title=",[b,c">,[b,c+(-1)^[b,[c,a+(-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 In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...

of a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

is a Gerstenhaber algebra. *The differential forms on a

Poisson manifold In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hami ...

form a Gerstenhaber algebra. *The multivector fields on a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

form a Gerstenhaber algebra using the Schouten–Nijenhuis bracket

References

* * * * Algebras Theoretical physics Symplectic geometry {{algebra-stub