Solenoids in automorphism groups of evolution algebras

TL;DR

This paper characterizes the automorphism groups of evolution algebras, especially those that are diagonalizable or satisfy the 2LI condition, revealing structures like inverse limits and dyadic solenoids.

Contribution

It provides a detailed description of automorphism groups of evolution algebras, including cases with diagonalizable automorphisms and those satisfying the 2LI condition, connecting algebraic and graph-theoretic structures.

Findings

Automorphism group as inverse limit of a diagram from the associated graph.

In certain cases, the automorphism group can be realized as a dyadic solenoid.

Complete description of automorphisms for algebras satisfying the 2LI condition.

Abstract

Let A be an evolution algebra (possibly infinite-dimensional) equipped with a fixed natural basis B, and let E be the associated graph defined by Elduque and Labra. We describe the group of automorphisms of A that are diagonalizable with respect to B. This group arises as the inverse limit of a functor (a diagram) from the category associated with the graph E to the category of groups. In certain cases, this group can be realized as a dyadic solenoid. Additionally, we investigate the automorphisms that permute (and possibly scale) the elements of B. In particular, for algebras satisfying the 2LI condition, we provide a complete description of their automorphism group.