On the Structure of Bernstein Algebras

Bernstein algebras were introduced by Holgate [6] as an algebraic formulation of the problem of classifying the stationary evolution operators in genetics (see [8]). 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 sorre types of Bernstein algebras (for instance, Bernstein algebras of dimension less than or equal to four, see [12, 9, 2]) as well as some other structural results (see for instance [1, 5, 11, 4]). However, due to the intrinsic complexity of Bernstein algebras, there is not a general structure theory for these objects which may serve as a starting point for a general classification theorem. In this paper we contribute to such a theory through two main ideas, direct products and Bernstein-Jordan algebras. We define direct products of Bernstein algebras in Section 2, together with the related notions of decomposable and indecomposable algebras, and mention some of their properties, including a Krull-Schmidt Theorem of uniqueness of such decompositions. In Section 3 we deal with Bernstein-Jordan algebras. This subject has been considered by several authors, and we collect here some known results. We also show that being Jordan is not just an artificial condition for a Bernstein algebra. A class of Bernstein-Jordan algebras which we call reduced, arise naturally as homomorphic images of Bernstein algebras, and in fact, a functor from the category of Bernstein algebras into the category of Bernstein-Jordan algebras can be defined. Sections 2 and 3 suggest the following approach to the structure of Bernstein algebras: the first step is to take a quotient algebra which is reduced, then to decompose the resulting algebra as a direct sum of indecomposable algebras and finally to classify the indecomposable Bernstein-Jordan algebras obtained in that way. In Section 4 we show that an obstacle to this approach appears at the last step. We give examples of infinite families of indecomposable Bernstein-Jordan algebras thus showing that the classification of these algebras presents some serious difficulties.