A note on cohomology of Clifford algebras

TL;DR

This paper introduces Clifford cohomology, a new cohomology theory for complex Clifford algebras, which helps classify their deformations and Morita equivalence, and extends to smooth sections over manifolds with Spin^c structures.

Contribution

It constructs Clifford cohomology for complex Clifford algebras and demonstrates its role in deformation classification and Morita equivalence, extending to bundles over manifolds.

Findings

Clifford cohomology controls algebra deformations.

It classifies complex Clifford algebras up to Morita equivalence.

Hochschild cohomology of Clifford algebra bundles studied.

Abstract

In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by {\it Clifford cohomology.} We show that {\it Clifford cohomology} controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold \(M\) which admits a \(Spin^{c}\) structure.