Termination for a class of algorithms for constructing algebras given by generators and relations

Abstract

We consider a certain type of algorithm designed to construct the multiplication table of algebras given by generators and relations. These computations may be performed for various classes of (not necessarily associative) algebras, such as Lie (super)algebras, Jordan algebras, associative algebras; in general, for any class of algebras axiomatised by suitable polynomial identities. The type of algorithm considered is based on straightforward computations in the free non-associative algebra on the generators, which do not depend in an essential way on the axioms of the class of algebras, in particular no concept of “representation” of the algebra is used. The algorithm itself is only partially specified: it proceeds by repeatedly taking steps chosen from a limited repertoire of very simple possibilities, but no fixed strategy for selecting steps is assumed. We study the question whether termination of the algorithm is guaranteed for those inputs that actually describe a finite-dimensional algebra (the question whether some arbitrary input has this property is not algorithmically decidable). We prove under certain assumptions about the strategy that termination is indeed guaranteed in this case.