Simple finite-dimensional algebras without finite basis of identities

Abstract

In 1993, Shestakov posed a problem of existence of a central simple finite-dimensional algebra over a field of characteristic 0 whose identities cannot be defined by a finite set (Dniester Notebook, Problem 3.103). In 2012, Isaev and the author constructed an example that gave a positive answer to this problem. In 2015, the author constructed an example of a central simple seven-dimensional commutative algebra without finite basis of identities. In this article we continue the study of Shestakov’s problem in the case of anticommutative algebras. We construct an example of a simple seven-dimensional anticommutative algebra over a field of characteristic 0 without finite basis of identities.