Hopf algebras and associative representations of two-dimensional evolution algebras

TL;DR

This paper explores the relationship between 2-dimensional evolution algebras and Hopf algebras, focusing on automorphism groups, associative representations, and their algebraic structures, with implications for finite-dimensional cases.

Contribution

It establishes a connection between evolution algebras and Hopf algebras, detailing the structure of associated algebras and automorphism groups, and characterizing when faithful associative representations exist.

Findings

Automorphism group of A is trivial iff no faithful associative representation exists.

Existence of faithful associative representation depends on Hopf algebra structure and characteristic of the field.

For perfect A with a faithful tight p-algebra, the p-algebra is isomorphic to the associated Hopf algebra.

Abstract

In this paper, we establish a connection between evolution algebras of dimension two and Hopf algebras, via the algebraic group of automorphisms of an evolution algebra. Initially, we describe the Hopf algebra associated with the automorphism group of a 2-dimensional evolution algebra. Subsequently, for a 2-dimensional evolution algebra $A$ over a field $K$, we detail the relation between the algebra associated with the (tight) universal associative and commutative representation of $A$, referred to as the (tight) $p$-algebra, and the corresponding Hopf algebra, $\mathcal{H}$, representing the affine group scheme $\text{Aut}(A)$. Our analysis involves the computation of the (tight) $p-$algebra associated with any 2-dimensional evolution algebra, whenever it exists. We find that $\text{Aut}(A)=1$ if and only if there is no faithful associative and commutative representation for $A$. Moreover, there is a faithful associative and commutative representation for $A$ if and only if $\mathcal{H}\not\cong K$ and $\text{char} (K)\neq 2$, or $\mathcal{H}\not\cong K(\epsilon)$ (the dual numbers algebra) and $\mathcal{H}\not\cong K$ in case of $\text{char} (K)= 2$. Furthermore, if $A$ is perfect and has a faithful tight $p$-algebra, then this $p$-algebra is isomorphic to $\mathcal{H}$ (as algebras). Finally, we derive implications for arbitrary finite-dimensional evolution algebras.