Duplicate, Bernstein algebras and evolution algebras
Abstract
In this paper, we firstly study a commutative algebra E over a field F of Char(F) ̸= 2 that satisfying dim(E2) = 1. We show that, such an algebra is an evolution algebra. Afterwards, we pay attention to commutative duplicate of a commutative algebra E. We find necessary and sufficient condition in which the duplicate D(E) is an evolution algebra. And, we finish by studying an evolution algebra that is a Bernstein algebra. We classify that algebras, up to isomorphism, in dimension ≤ 4.
{\bf Keywords}: Evolution algebras, Bernstein algebras, Duplicate, natural base.
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