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 natural phenomena. This is in contrast to experimental physics, which uses experi ...

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

algebraic structure In mathematics, an algebraic structure 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 multiplication), and a finite set ...

discovered by

Murray Gerstenhaber Murray Gerstenhaber (born June 5, 1927) is an American mathematician and professor of mathematics at the University of Pennsylvania, best known for his contributions to theoretical physics with his discovery of Gerstenhaber algebra. He is also a ...

(1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. 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, whose ...

. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory 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. T ...

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 identi ...

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. ...

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''">'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 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, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

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 iden ...

is a Gerstenhaber algebra. *The differential forms on a

Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalen ...

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 ...

form a Gerstenhaber algebra using the

Schouten–Nijenhuis bracket In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two differe ...

References

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