nLab Gerstenhaber algebra

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Contents

Contents

Idea

A Gerstenhaber algebra is a Poisson 2-algebra, hence a graded Poisson algebra (i.e. internal to graded vector spaces) with Poisson bracket of degree -1 (cf. Cattaneo, Fiorenza & Longoni 2006, Def. 1.1).

Since the signs in the Jacobi identity depend only on the degree of the bracket modulo 2, some authors speak more generally of Gerstenhaber algebras in the case of graded Poisson algebras with bracket of any odd degree (e.g. Kontsevich 1999, Thm. 3).

Definition

Definition

A Gerstenhaber algebra is a chain complex AA equipped with

  1. a graded symmetric product ()():AAA(-)\cdot(-) \colon A \otimes A \to A,

  2. a graded skew-symmetric bracket [,]:AAA[1][-,-] \;\colon\; A \otimes A \to A[1],

such that

Properties

Theorem

The homology of the operad for Gerstenhaber algebras in chain complexes is the operad for Gerstenhaber algebras.

Accordingly the homology of an E2-algebra is a Gerstenhaber algebra.

This is due to Cohen (1976).

Remark

A Gerstenhaber algebra equipped in addition with a certain morphism Δ:AA\Delta : A \to A is a BV-algebra. This is the homology of an algebra over the framed little 2-disk operad.

References

Last revised on June 25, 2024 at 17:06:55. See the history of this page for a list of all contributions to it.