${\mathcal O}$-operators of Loday algebras and analogues of the classical Yang-Baxter equation

TL;DR

This paper introduces ${\mathcal O}$-operators for Loday algebras, generalizing Rota-Baxter operators, and explores their relation to analogues of the classical Yang-Baxter equation, revealing structural conditions and invariances.

Contribution

It defines ${\mathcal O}$-operators for Loday algebras, connects them to Yang-Baxter analogues, and characterizes algebraic structures via invertible operators and bilinear form invariances.

Findings

Invertible ${\mathcal O}$-operators characterize algebraic operation expansions.

Analogues of the classical Yang-Baxter equation are realized as ${\mathcal O}$-operators.

Constraints on bilinear forms are established for these algebras.

Abstract

We introduce notions of ${\mathcal O}$-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota-Baxter operators. The invertible $\mathcal O$-operators give a sufficient and necessary condition on the existence of the $2^{n+1}$ operations on an algebra with the $2^{n}$ operations in an associative cluster. The analogues of the classical Yang-Baxter equation in these algebras can be understood as the $\mathcal O$-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.