Tensor product of evolution algebras

TL;DR

This paper explores the properties of evolution algebras under tensor products, providing a complete classification of four-dimensional perfect evolution algebras as tensor products of two-dimensional ones.

Contribution

It demonstrates that the class of evolution algebras is closed under tensor products and classifies four-dimensional perfect evolution algebras arising from two-dimensional factors.

Findings

Four-dimensional evolution algebras can be classified as tensor products of two-dimensional algebras.

Some four-dimensional evolution algebras are tensor products of non-evolution algebras.

Inheritance of properties like nondegeneracy and simplicity under tensor products is analyzed.

Abstract

The starting point of this work is that the class of evolution algebras over a fixed field is closed under tensor product. This arises questions about the inheritance of properties from the tensor product to the factors and conversely. For instance nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures so we achieve a complete classification of four-dimensional perfect evolution algebras arising as tensor product of two-dimensional ones. We find that there are 4-dimensional evolution algebras which are the tensor product of two nonevolution algebras.