Linear deformations of discrete groups and constructions of multivalued groups

Abstract

We construct deformations of discrete multivalued groups described as special deformations of their group algebras in the class of finite-dimensional associative algebras. We show that the deformations of ordinary groups producing multivalued groups are defined by cocycles with coefficients in the group algebra of the original group and obtain classification theorems on these deformations. We indicate a connection between the linear deformations of discrete groups introduced in this paper and the well-known constructions of multivalued groups. We describe the manifold of three-dimensional associative commutative algebras with identity element, fixed basis, and a constant number of values. The group algebras of -valued groups of order three (three-dimensional -group algebras) form a discrete set in this manifold.