Construction of some non-associative algebras from Associative Algebras with an endomorphism operator, a differential operator or a left averaging operator

TL;DR

This paper explores how various linear operators on associative algebras can be used to construct different types of non-associative algebras, providing explicit examples on matrix subspaces.

Contribution

It introduces the construction of non-associative algebras from associative algebras using endomorphism, differential, and averaging operators, with concrete matrix examples.

Findings

Constructed non-associative algebras from associative algebras using linear operators.

Provided explicit examples on subspaces of 3x3 real matrices.

Demonstrated the induction of Lie, Pre-Lie, Jordan, and Leibniz algebra structures.

Abstract

In this paper, we introduce the concepts of endomorphism operator, left averaging operator, differential operator and Rota-Baxter Operator, and we construct examples of these linear maps on associative algebras with a left identity, a skew-idempotent or an idempotent element. These maps on associative algebra induce a non-associative algebra structure such as Lie algebra, Pre-Lie algebra, Jordan algebra, Flexible Algebra or (left) Leibniz algebra. We consider the construction of non-associative algebras from associative algebras with Linear Operators as the main results of this work. In this paper we give examples of non-associative algebras on subspaces of square matrices M3(R).