A note on Bernstein algebras
type: Item
instance of: scholarly article
title: A note on Bernstein algebras
zbMATH Open document ID: 0673.17024
DOI: 10.1016/0024-3795(89)90595-8
author: Roberto Costa
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17D92
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 17A30
zbMATH DE Number: 4101434
zbMATH Keywords: structure
zbMATH Keywords: genetic algebras
zbMATH Keywords: Bernstein algebras
zbMATH Keywords: baric algebra
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90595-8
OpenAlex ID: W2016203242
cites work: Q3705576
cites work: Q5628309
cites work: Genetic Algebras Satisfying Bernstein's Stationarity Principle
cites work: Q3293461
cites work: Q5590138
cites work: Q3869125
cites work: Bernstein algebras
cites work: Q3889234
description: scientific article; zbMATH DE number 4101434