Periodic algebras generated by groups

TL;DR

This paper studies algebras generated by group elements with multiplication defined via a function, exploring conditions for Leibniz structure, periodicity, and nilpotency, including classifications for integer groups.

Contribution

It introduces a framework for periodic group-based algebras, establishing criteria for Leibniz properties and nilpotency, and classifies certain periodic algebras over integers.

Findings

Condition for algebra to be Leibniz algebra

Criterion for right nilpotency of periodic algebras

Classification of 2Z- and 3Z-periodic algebras over integers

Abstract

We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $G\times G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In particular, we find a condition on $f$ under which the corresponding algebra is a Leibniz algebra. Moreover, for a given subgroup $\hat G$ of $G$ we define a $\hat G$-periodic algebra, which corresponds to a $\hat G$-periodic function $f,$ we establish a criterion for the right nilpotency of a $\hat G$-periodic algebra. In addition, for $G=\mathbb Z$ we describe all $2\mathbb Z$- and $3\mathbb Z$-periodic algebras. Some properties of $n\mathbb Z$-periodic algebras are obtained.