Some properties of the automorphisms of a Bernstein algebra
Bernstein algebras had their origin as a formal framework for the Hardy-Weinberg law of classical population genetics. These algebras have been studied as purely algebraic objects, with no regard to applications in genetics. A baric algebra \((A,\omega)\) is a commutative algebra \(A\) over a field \(K\) of characteristic not 2 with a weight homomorphism \(\omega:A \rightarrow K\), that is a nonzero algebra homomorphism. A Bernstein algebra over \(K\) is a commutative nonassociative baric algebra \((A,\omega)\) satisfying \((x^2)^2=\omega (x)^2x^2\) for all \(x \in A\). Bernstein algebras have idempotents \(e\) such that \(\omega (e)=1\). If \(e\) is an idempotent element of a Bernstein algebra \(A\), \(e\) determines a decomposition \(A= Fe\oplus U_e\oplus V_e\) called the Peirce decomposition of \(A\), where ker\((\omega)=U_e\oplus V_e\). If \(A^2=0\) then \(A\) is called nuclear. In this case \(V_e^2=U_e\). The study of the multiplication algebra \(M(A)\) of a Bernstein algebra \(A\) has been developed by several authors. In the present work, the authors show that if \(A\) is a nuclear Bernstein algebra then the group of automorphisms of \(M(A)\) has a proper subgroup isomorphic to \(\operatorname {Aut} A\).
scientific article; zbMATH DE number 1997869