Malcev Yang-Baxter equation, weighted $\mathcal{O}$-operators on Malcev algebras and post-Malcev algebras

TL;DR

This paper explores $ ext{O}$-operators on Malcev algebras, introduces weighted variants and post-Malcev algebras, and connects these structures to solutions of the Malcev Yang-Baxter equation and new algebraic frameworks.

Contribution

It introduces weighted $ ext{O}$-operators and post-Malcev algebras, expanding the algebraic tools for Malcev algebras and their relation to the Yang-Baxter equation.

Findings

Characterization of weighted $ ext{O}$-operators via semi-direct product graphs

Introduction of post-Malcev algebras as underlying structures

Establishment of connections between post-Malcev and Malcev algebras

Abstract

The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra. Then we introduce a new algebraic structure called post-Malcev algebras. Therefore, post-Malcev algebras can be viewed as the underlying algebraic structures of weighted $\mathcal{O}$-operators on Malcev algebras. A post-Malcev algebra also gives rise to a new Malcev algebra. Post-Malcev algebras are analogues for Malcev algebras of post-Lie algebras and fit into a bigger framework with a close relationship with post-alternative algebras.