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. 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 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.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (2)

Related concepts (6)

Related lectures (1)

Gerstenhaber structure on Hochschild cohomology of the Fomin–Kirillov algebra on 3 generators

The goal of this article is to compute the Gerstenhaber bracket of the Hochschild cohomology of the Fomin–Kirillov algebra on three generators over a field of characteristic different from 2 and 3. This is in part based on a general method we introduce to ...

2022

Extensions of Lie-Rinehart algebras and cotangent bundle reduction

Tudor Ratiu

Let Q denote a smooth manifold acted upon smoothly by a Lie group G. The G-action lifts to an action on the total space TQ of the cotangent bundle of Q and hence on the standard symplectic Poisson algebra of smooth functions on TQ. The Poisson algebra of ...

Oxford University Press2013

Poisson algebra

In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.

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. The prefix super- comes from the theory of supersymmetry in theoretical physics. Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra.

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. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems.

Hamiltonian Mechanics: Phase Space and Poisson Bracket CH-453: Molecular quantum dynamics

Explores phase space, the Poisson bracket, and Hamiltonian mechanics concepts.