Products of commutator ideals of some Lie-admissible algebras

TL;DR

This paper investigates the properties of commutator ideals in various Lie-admissible algebras, establishing nilpotency results and analyzing the structure of their lower central chains.

Contribution

It provides new insights into the behavior of commutator ideals in Novikov, bicommutative, and assosymmetric algebras, including nilpotency and structural properties.

Findings

Commutator ideals in Lie nilpotent Novikov and bicommutative algebras are nilpotent.

Properties of lower central chains are characterized for these algebras.

Products of commutator ideals in assosymmetric algebras exhibit properties similar to associative algebras.

Abstract

In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central chains for Novikov algebras and bicommutative algebras. Then we show that for every Lie nilpotent Novikov algebra or Lie nilpotent bicommutative algebra $\mathcal{A}$, the ideal of $\mathcal{A}$ generated by the set $\{ab - ba\mid a, b\in \mathcal{A}\}$ is nilpotent. Finally, we study properties of the lower central chains for assosymmetric algebras, study the products of commutator ideals of assosymmetric algebras and show that the products of commutator ideals have a similar property as that for associative algebras.