A Geometric Approach to Hochschild Cohomology of the Exterior Algebra

TL;DR

This paper introduces a new method for computing Hochschild (co)homology of exterior algebras, revealing its structure and formal properties, with implications for deformation theory and formality conjectures.

Contribution

It provides a novel geometric approach to Hochschild cohomology of exterior algebras, including explicit computations and analysis of formality conditions.

Findings

Hochschild cohomology corresponds to even-weight polyvector fields.

Formality holds when the generating vector space is even-dimensional.

Conjecture that formality fails in odd dimensions, proven for dimension one.

Abstract

We give a new computation of Hochschild (co)homology of the exterior algebra, together with algebraic structures, by direct comparison with the symmetric algebra. The Hochschild cohomology is determined to be essentially the algebra of even-weight polyvector fields. From Kontsevich's formality theorem, the differential graded Lie algebra of Hochschild cochains is proved to be formal when the vector space generating the exterior algebra is even dimensional. We conjecture that formality fails in the odd dimensional case, proving this when the dimension is one. In all dimensions, formal deformations of the exterior algebra are classified by formal Poisson structures.