Generic 8-dimensional algebras with mixed basis-graph

Thierry Dana-Picard

Vol. 164 (1994), No. 2, 229–261

DOI: 10.2140/pjm.1994.164.229

Abstract

Deformation theory is the appropriate tool for describing the irreducible components of the scheme Alg _n which parametrizes the structures of n-dimensional associative algebras with unit. Each component is “dominated” by one generic or quasi-generic algebra or family of algebras (genericity means that the algebra or the family has only trivial infinitesimal deformations, and quasi-genericity means that the algebra or the family has non trivial infinitesimal deformations, but no algebraic deformation). The components dominated by a generic algebra (or family) are reduced, while the components dominated by a quasi-generic family are non reduced. The invariants we use for that classification are the basis-graph, both weighted and unweighted, of an associative algebra. In this paper, we classify the 8-dimensional algebras with mixed basis-graph and give lower bounds for the numbers of irreducible components of the scheme Alg ₈, reduced and non reduced.

Mathematical Subject Classification 2000

Primary: 16S80

Milestones

Received: 3 February 1992

Revised: 15 September 1992

Published: 1 June 1994