On a class of Algebras Satisfying polynomial identity of degree six

TL;DR

This paper investigates the structure of a specific class of algebras satisfying a degree-six polynomial identity, revealing their decomposition properties and connections to various algebraic systems.

Contribution

It establishes the existence of Peirce decomposition in such algebras assuming a non-zero idempotent, and explores their links with Bernstein, Jordan, and train algebras.

Findings

Algebras admit Peirce decomposition with a non-zero idempotent.

Connections established with Bernstein, Jordan, and power associative algebras.

Structural insights into polynomial identity of degree six.

Abstract

In this paper we study the structure of a class of algebras satisfying a polynomial identity of degree 6. We show, assuming the existence of a non-zero idempotent, that if an algebra satisfies such an identity, it admits a Peirce decomposition related to this idempotent. We studied the algebraic structure and highlighted the connections of the algebras of this class with Bernstein algebras, train algebras, Jordan algebras and power associative algebras. Keywords: Peirce decomposition, Bernstein algebra, Jordan algebra, Power associative algebra, train algebra,polynomial identity, idempotent.