nLab Gerstenhaber algebra

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 $A$ equipped with

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

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

such that

• $(-) \cdot (-)$ is associative

• $[-,-]$ satisfies the Jacobi identity

• $[a,-]$ is a derivation of $(-)\cdot(-)$ for all $a \in A$.

Properties

Theorem

The homology of the little 2-disk operad in chain complexes is the operad for Gerstenhaber algebras.

Accordingly, the homology of an $E_2$-algebra is a Gerstenhaber algebra.

This is due to Cohen 1976.

Remark

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

Related concepts

• Poisson n-algebra

• BV-algebra

References

• Fred Cohen, The homology of $C_{n+1}$-Spaces, $n\geq0$, In: The Homology of Iterated Loop Spaces. Lecture Notes in Mathematics, vol. 533. Springer, Berlin, Heidelberg. (1976) [doi:10.1007/BFb0080467]

• Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963) 267-288 [doi:10.2307/1970343, jstor:1970343 MR28:5102]

• Ping Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys. 200 3 (1999) 545-560 [arXiv:dg-ga/9703001, doi:10.1007/s002200050540]

• Maxim Kontsevich, Operads and Motives in Deformation Quantization, Lett. Math. Phys. 48 (1999) 35-72 [arXiv:math/9904055, doi:10.1023/A:1007555725247]

• Alberto S. Cattaneo, Domenico Fiorenza, Riccardo Longoni, Graded Poisson Algebras, in: Encyclopedia of Mathematical Physics, Elsevier (2006) 560-567 [arXiv:1811.07395, doi:10.1016/B0-12-512666-2/00434-X]