Solenoids in automorphism groups of evolution algebras
Yolanda Cabrera Casado, Maria Inez Cardoso Gonçalves, Daniel Gonçalves, Dolores Martín Barquero, Cándido Martín González, Iván Ruiz Campos
TL;DR
This work characterizes automorphism groups of evolution algebras with a fixed natural basis through a graph-theoretic lens, showing that diagonalizable automorphisms form an inverse limit of a diagram tied to the graph and can realize as a dyadic solenoid in certain cases. It also develops a framework for basis-permuting and scaling automorphisms, proving basis-independence under the 2LI condition and providing a semidirect-product description when the structure matrix is invertible or when graph symmetries are present. The paper introduces weighted-graph techniques to treat non-diagonalizable automorphisms, proving category equivalences that yield inverse-limit descriptions and practical computation methods. Collectively, these results deepen the understanding of the symmetry and automorphism structure of both finite and infinite-dimensional evolution algebras, with solenoidal and Tate-module-like phenomena emerging in natural settings.
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.