Low Dimensional Bernstein-Jordan Algebras

Abstract

Bernstein algebras were introduced by Holgate [5] as an algebraic formulation of the problem of classifying the stationary evolution operators in genetics (see [7]). Since then, many authors have contributed to the study of these algebras, and there is a fairly extensive bibliography on the subject. Known results include classification theorems for some types of Bernstein algebras (for instance, Bernstein algebras of dimension less than or equal to four, see [10, 8, 2]) as well as some other structural results (see for instance [1, 5, 9, 4]). In [3], the authors suggested an approach to the study of the structure of Bernstein algebras through two main ideas: direct products and the related concepts of decomposability and indecomposability, and the reduction of the general problem to the study of Bernstein-Jordan algebras. The procedure depended on getting what was called a reduced Bernstein algebra (which is, in particular, a Jordan algebra), and on the description of the indecomposable factors of the algebra so obtained. In this paper we use these ideas to classify Bernstein-Jordan algebras of low dimension. Thus, in Section 3 we describe reduced Bernstein algebras of dimension less than or equal to 5 through their indecomposable factors. We use this information in Section 4 to recover all Bernstein-Jordan algebras of dimension less than or equal to 5 by gluing together reduced algebras and the trivial ideal that, when factored out, creates reduced algebras. After a Section 1 of preliminaries, where we collect several known facts about Bernstein algebras which will be used later, we devote Section 2 to proving some technical lemmas which will ease the computations of Section 3.