Processing identities by group representation

TLDR: In this paper, the authors present a technique that enables a computer to process identities in non-associative algebras, where identities are expressed as elements of the group ring on the symmetric group and then transferred to matrices by the well-known isomorphisms of group rings.

Abstract: Publisher Summary This chapter presents a technique that enables a computer to process identities. The examples used are from nonassociative algebras, but the method is general and can be applied to many other situations. Identities are expressed as elements of the group ring on the symmetric group and then are transferred to matrices by the well-known isomorphisms of group rings. This approach has several advantages over the standard approach, which uses a system of linear equations. The system indicates whether a desirable identity is a consequence of other assumed identities. If the identity is not a consequence of the assumed identities, it tells how close it is to being true. This lets the investigator know the routes that are worth pursuing. Second, it uses very little array area. Identities involving four unknowns are processed, with the largest array being a 3 × 3 matrix.