On commutative left-nilalgebras of index 4

TL;DR

This paper proves that in certain commutative nonassociative algebras satisfying a specific identity, high-order products of left multiplication operators vanish, confirming a conjecture and advancing understanding of algebraic identities.

Contribution

It provides a positive solution to a conjecture regarding identities in commutative nonassociative algebras of index 4, specifically proving the vanishing of certain high-order operator products.

Findings

Confirmed the conjecture for characteristic not 2 or 3.

Established that products of left multiplication operators vanish when the sum of exponents is at least 10.

Extended the understanding of identities in commutative nonassociative algebras.

Abstract

We first present a solution to a conjecture of I. Correa, A. Labra and I.R. Hentzel in the positive. We prove that if $A$ is a commutative nonassociative algebra over a field of characteristic $\ne 2,3$, satisfying the identity $x(x(xx))=0$, then $L_{a^{t_1}}L_{a^{t_2}}\cdots L_{a^{t_s}} \equiv 0$ if $ t_1+t_2+\cdots + t_s \geq 10$, where $a\in A$.