A note on Bernstein algebras
Bernstein algebras have been defined by \textit{P. Holgate} in 1975 after Lyubich had investigated in great detail the problem posed by Bernstein of the condition required to ensure that a population attains equilibrium after one generation. A commutative nonassociative algebra A over a field F endowed with a nonzero homomorphism w: \(A\to F\) is called a baric algebra and w is called the weight function of A. We say A is a Bernstein algebra if the identity \((x^ 2)^ 2=w(x)^ 2x^ 2\) holds in A. The kernel of w, called also the kernel of A is an ideal of codimension 1 satisfying the identity \((x^ 2)^ 2=0.\) The purpose of this paper is to study the following purely algebraic problem: given a commutative finite dimensional algebra B over F (char \(F\neq 2)\) such that \((x^ 2)^ 2=0\), which are the Bernstein algebras \(B'\) having B as kernel. Then \(B'\) must be the direct sum of B and F with the following multiplication rule: \((x,a)(y,b)=(xy, ab+t(xb+ya))\) with x,y in F, a,b in B and t: \(B\to B\) a linear mapping such that: \(a^ 2t(a)=0\) for all a in B, \(a^ 2=4t(a)^ 2+2t(a^ 2)\) for all a in B and \(t=2t^ 2\). Thus the problem of classifying Bernstein algebras is reduced to that of classifying all algebras satisfying the identity \((x^ 2)^ 2=0\) and then solve the problem of the embedding. The paper deals with the case dim \(B^ 2=1\) when, if \(B^ 2=Fc\) and \(xy=b(x,y)c\) one can associate the bilinear form b, which has an isotropic vector, and then the author uses this correspondence to describe classes of isomorphism of algebras (Theorem 1). The case when the bilinear form b(x,y) has Witt index 0 is dealt with in Theorem 2 which asserts that B can be embedded in a finite number of nonisomorphic Bernstein algebras described by two explicitly given Tables.
scientific article; zbMATH DE number 4101434