FINITE SPECTRAL SEQUENCES AND MASSEY POWERS IN THE DEFORMATION-THEORYOF GRADED LIE-ALGEBRAS AND ASSOCIATIVE ALGEBRAS

A 'deformation' of an algebra G with multiplication mu is a non-isomor phic multiplication mu' on the same underlying vector space that is 'i nfinitesimally close' to mu. 'Deformation theory' is the attempt to cl assify such deformations. In the category of graded algebras one is pr imarily interested in deformations that produce filtered algebras. In this paper we propose the study of finite spectral sequences in this t heory. We show that an 'order-n-deformation' of a graded Lie algebra o r 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 o f determining whether or not two given deformations with the same infi nitesimals are equivalent. It also solves the problem of dependency of classical obstructions to extending an order-n-deformation to an orde r-(n + 1)-deformation on representatives chosen for the deformation.