The paper is devoted to the study of evolution algebras that are power-associative algebras. We give the Wedderburn decomposition of evolution algebras that are power-associative algebras and we prove that these algebras are Jordan algebras. Finally, we use this decomposition to classify these algebras up to dimension six.
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.
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