Evolution algebras of arbitrary dimension and their decompositions

TL;DR

This paper provides a comprehensive analysis of evolution algebras of any dimension, focusing on their structure, decomposition, and the role of associated graphs, especially distinguishing between degenerate and non-degenerate cases.

Contribution

It introduces a detailed framework for understanding evolution subalgebras, ideals, and decompositions, including a novel fragmentation process for finite-dimensional cases.

Findings

Non-degenerate evolution algebras are irreducible iff their associated graph is connected.

Every non-degenerate evolution algebra admits a unique decomposition into irreducible components.

The associated graph's structure is fundamental in characterizing the algebra's properties.

Abstract

We study evolution algebras of arbitrary dimension. We analyze in deep the notions of evolution subalgebras, ideals and non-degeneracy and describe the ideals generated by one element and characterize the simple evolution algebras. We also prove the existence and unicity of a direct sum decomposition into irreducible components for every non-degenerate evolution algebra. When the algebra is degenerate, the uniqueness cannot be assured. The graph associated to an evolution algebra (relative to a natural basis) will play a fundamental role to describe the structure of the algebra. Concretely, a non-degenerate evolution algebra is irreducible if and only if the graph is connected. Moreover, when the evolution algebra is finite-dimensional, we give a process (called the fragmentation process) to decompose the algebra into irreducible components.