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

In this paper we introduce an analog of the (classical) YangBaxter equation (CYBE) for vertex operator algebras (VOAs) in its tensor form, called the vertex operator Yang-Baxter equation (VOYBE). Specialized to degree-one subspace 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 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 Tr : U' -> 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 Ware characterized by the condition that their skewsymmetrizations are solutions to the 0-VOYBE in the semidirect product VOA V W'. Specialized to the degree-one subspace 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. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.