Classical Yang-Baxter equation for vertex operator algebras and its operator forms

TL;DR

This paper introduces a vertex operator algebra analog of the classical Yang-Baxter equation, called the VOYBE, and explores its operator form and solutions in relation to relative Rota-Baxter operators, extending Lie algebra concepts.

Contribution

It defines the VOYBE for VOAs, introduces relative RBOs in this context, and establishes their correspondence with solutions to the VOYBE, generalizing classical Lie algebra results.

Findings

VOYBE reduces to CYBE at level one of a VOA.

Skewsymmetric solutions to VOYBE correspond to relative RBOs.

Strong relative RBOs relate to solutions of the 0-VOYBE in semidirect product VOAs.

Abstract

In this paper we introduce an analog of the (classical) Yang-Baxter equation (CYBE) for vertex operator algebras (VOAs) in its tensor form, called the vertex operator Yang-Baxter equation (VOYBE). When specialized to level one of a vertex operator algebra, the VOYBE reduces to the CYBE for Lie algebras. To give an operator form of the VOYBE, we also introduce the notion of relative Rota-Baxter operators (RBOs) as the VOA analog of relative RBOs (classically called $\mathcal{O}$-operators) for Lie algebras. It is shown that skewsymmetric solutions $r$ to the VOYBE in a VOA $U$ are characterized by the condition that their corresponding linear maps $T_r:U'\to U$ from the graded dual $U'$ of $U$ are relative RBOs. On the other hand, strong relative RBOs on a VOA $V$ associated to an ordinary $V$-module $W$ are characterized by the condition that their antisymmetrizers are solutions to the $0$-VOYBE in the semidirect product VOA $V\rtimes W'$. Specializing to the first level of a VOA, these relations between the solutions of the VOYBE and the relative RBOs for VOAs recover the classical relations between the solutions of the CYBE and the relative RBOs for Lie algebras.