$\mathcal{O}$-operators and related structures on Leibniz algebras

TL;DR

This paper explores advanced algebraic structures related to $\\mathcal{O}$-operators on Leibniz algebras, introducing new structures and revealing their interconnections through solutions of Maurer-Cartan equations.

Contribution

It introduces (dual) $\mathcal{O}$N-structures, studies their relations with $\mathcal{O}$-operators, and connects solutions of Maurer-Cartan equations to these structures on Leibniz algebras.

Findings

$\\mathcal{O}$-operators and dual $\ ext{O}$N-structures generate each other under certain conditions.

Solutions to the strong Maurer-Cartan equation induce dual $\ ext{O}$N-structures.

Interrelations among $r$-n, RBN-, and $\ ext{BN}$-structures are established.

Abstract

An $\mathcal{O}$-operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to $\mathcal{O}$-operators on Leibniz algebras and introduce (dual) $\mathcal{O}$N-structures on Leibniz algebras associated to their representations. It is proved that $\mathcal{O}$-operators and dual $\mathcal{O}$N-structures generate each other under certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual $\mathcal{O}$N-structure. Finally, $r-n$ structures, RBN-structures and $\mathcal{B}N$-structures on Leibniz algebras are thoroughly studied and their interdependent relations are also studied.