Regular evolution algebras are universally finite

Abstract

In this paper we show that evolution algebras over any given field k \Bbbk are universally finite. In other words, given any finite group G G , there exist infinitely many regular evolution algebras X X such that A u t ( X ) ≅ G Aut(X)\cong G . The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme G \mathbf {G} over k \Bbbk is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra.

Automorphism groups of Cayley evolution algebras

“…We end this section by pointing out that our main result is an improvement of our previous result [5,Theorem 1.1] where finite groups were realised by regular, but not simple (see Remark 3.6), evolution algebras (see also [15] for an independent proof in char k = 0).…”

Automorphism groups of Cayley evolution algebras

“…We end this section by pointing out that our main result is an improvement of our previous result [3,Theorem 1.1] where finite groups were realised by regular, but not simple (see Remark 3.6), evolution algebras (see also [12] for an independent proof in char k = 0).…”

On Automorphism Groups of Idempotent Evolution Algebras

“…of AMS in May 2019, since the editor was unable to find a referee for the paper for over two years, it was then withdrawn from PAMS upon the suggestion of the editor and submitted to LAA. After it was accepted by LAA and subsequently posted on arXiv, the authors were informed the existence of the reference [6] We denote by E(A) the evolution algebra with the structure matrix A if we need to specify A, and denote by Γ A (or Γ E ) the graph whose adjacency matrix is obtained from A by replacing all nonzero entries of A by 1. The vertices of Γ A will be just e 1 , ..., e n .…”