Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras

TL;DR

This paper introduces nonabelian generalized Lax pairs, extended O-operators, and an extended classical Yang-Baxter equation, revealing new structures in Lie bialgebras and PostLie algebras with explicit classifications.

Contribution

It generalizes classical Lax pairs and Yang-Baxter equations by defining nonabelian versions and explores their connections to PostLie algebras and Lie bialgebra structures.

Findings

Established relationship between extended O-operators and the extended classical Yang-Baxter equation.

Provided explicit descriptions of Manin triples for new Lie bialgebra classes.

Linked PostLie algebra structures to nonabelian generalized Lax pairs.

Abstract

We generalize the classical study of (generalized) Lax pairs and the related $O$-operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended $\calo$-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized $r$-matrix ansatz and the related double Lie algebra structures. Relationship between extended $O$-operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind $O$-operators and fits in a setup of triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.