# Genetic Volterra algebras and their derivations

## TL;DR

This paper investigates the structure of genetic Volterra algebras, characterizing associative cases and derivations, and establishing conditions for trivial derivations in non-associative cases, especially in three dimensions.

## Contribution

It provides a complete description of associative genetic Volterra algebras and characterizes derivations, including a classification of three-dimensional cases and conditions for trivial derivations.

## Key findings

- All derivations are trivial in associative cases.
- A sufficient condition for trivial derivations in non-associative cases.
- All local derivations in three-dimensional cases are actual derivations.

## Abstract

The present paper is devoted to genetic Volterra algebras. We first study characters of such algebras. We fully describe associative genetic Volterra algebras, in this case all derivations are trivial. In general setting, i.e. when the algebra is not associative, we provide a sufficient condition to get trivial derivation on generic Volterra algebras. Furthermore, we describe all derivations of three dimensional generic Volterra algebras, which allowed us to prove that any local derivation is a derivation of the algebra.