In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.
This is joint work with C. Costoya (U. Santiago Compostela) and Pedro Mayorga (U. Malaga).
This video is part of the Non-Associative Day in Online, run by the European Non-Associative Algebra Seminar series.
