Hochschild DGLAs and torsion algebras

Abstract

The associator of a non-associative algebra is the curvature of the Hochschild quasi-complex. The relationship ``curvature-associator'' is investigated. Based on this generic example, we extend the geometric language of vector fields to a purely algebraic setting, similar to the context of Gerstenhaber algebras. We interprete the elements of a non-associative algebra with a Lie bracket as ``vector fields'' and the multiplication as a connection. We investigate conditions for the existance of an ``algebra of functions'' having as algebra of derivations the original non-associative algebra.