Evolution train algebras

Abstract

Through this paper, we show that the criteria for real evolution algebra to be a baric algebra can be extended to any evolution algebra over a commutative field of characteristic ≠2. Then we prove that an evolution algebra E is a train algebra of rank r + 1 if and only if the kernel of its weight function is nil of nil-index r > 1. We also study special train evolution algebra and characterize idempotents, power-associativity and automorphism in evolution train algebra. Finally we classify up to dimension 5, indecomposable evolution nil-algebra of nil-index 4 that are not power-associative.