Finite spectral sequences and massey powers in the deformation theory of graded lie algebras and associative algebras

Abstract

A ‘deformation’ of an algebra G with multiplication μ' is a non-isomorphic multiplication μ' on the same underlying vector space that is ‘infinitesimally close’ to μ. ‘Deformation theory’ is the attempt to classify such deformations. In the category of graded algebras one is primarily interested in deformations that produce filtered algebras. In this paper we propose the study of finite spectral sequences in this theory. We show that an ‘order- n-deformation’ of a graded Lie algebra or associative algebra G induces a finite spectral sequence, the first term of which is the corresponding Hochschild cohomology H ∗(G, G) . The study of this spectral sequence is necessary to resolve the problem of determining whether or not two given deformations with the same infinitesimals are equivalent. It also solves the problem of dependency of classical obstructions to extending an order- n-deformation to an order-( n + 1)-deformation on representatives chosen for the deformation.