Ternary mappings of some evolution algebras

TL;DR

This paper computes the group scheme of ternary automorphisms of evolution algebras, uses Lie functors to analyze ternary derivations, and classifies these derivations for finite-dimensional cases, including 2-dimensional algebras.

Contribution

It introduces a method to classify ternary derivations of evolution algebras using group schemes and Lie algebra techniques, providing explicit results for low-dimensional cases.

Findings

Computed the group scheme of ternary automorphisms for perfect finite-dimensional evolution algebras.

Applied the Lie functor to determine the Lie algebra of ternary derivations.

Classified all ternary derivations for 2-dimensional evolution algebras.

Abstract

The group scheme of ternary automorphisms of a perfect finite dimensional evolution algebra A is computed. The main advantage of using group schemes is that it allows to apply the Lie functor to determine the Lie algebra of ternary derivations of A. Using the generalised inverse of a matrix, we provide a precise classification of all ternary derivations of an arbitrary finite-dimensional evolution algebra A. The ternary derivations of all 2-dimensional evolution algebras are also computed.