Alg\`ebres et cog\`ebres de Gerstenhaber et cohomologie de Chevalley-Harrison

TL;DR

This paper explicitly describes the enveloping $G_infty$ algebra of a Gerstenhaber algebra, introduces the Chevalley-Harrison cohomology operator, and demonstrates its nontriviality for a natural subalgebra of polyvector fields.

Contribution

It provides a detailed description of the enveloping $G_infty$ algebra and defines the Chevalley-Harrison cohomology operator for Gerstenhaber algebras.

Findings

Explicit description of the enveloping $G_infty$ algebra.

Definition of the Chevalley-Harrison cohomology operator.

Proof of nontriviality of a cohomology group in a natural subalgebra.

Abstract

The fundamental example of Gerstenhaber algebra is the space $T_{poly}({\mathbb R}^d)$ of polyvector fields on $\mathbb{R}^d$, equipped with the wedge product and the Schouten bracket. In this paper, we explicitely describe what is the enveloping $G_\infty$ algebra of a Gerstenhaber algebra $\mathcal{G}$. This structure gives us a definition of the Chevalley-Harrison cohomology operator for $\mathcal{G}$. We finally show the nontriviality of a Chevalley-Harrison cohomology group for a natural Gerstenhaber subalgebra in $T_{poly}({\mathbb R}^d)$.