The isomorphism problem in the context of PI-theory for two-dimensional Jordan algebras

Abstract

Let F be a field of characteristic different from 2. Small-dimensional Jordan algebras over F have been extensively studied and classified. In the present paper we show that any two-dimensional Jordan algebras over a finite field are isomorphic if and only if they satisfy the same polynomial identities (the opposite happens in the case F is infinite, even if algebraically closed). We determine a finite generating set for the T-ideal of the polynomial identities of every two-dimensional Jordan algebra when F is finite, and linear bases for the corresponding relatively free algebras are also determined.