Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces

TL;DR

This paper explores how reflections on quadratic and relative Rota-Baxter Lie algebras generate solutions to the classical reflection equation and constructs Poisson Lie groups and homogeneous spaces from these algebraic structures.

Contribution

It introduces the concept of reflections on Rota-Baxter Lie algebras and links them to solutions of the classical reflection equation and Poisson geometric structures.

Findings

Reflections on quadratic Rota-Baxter Lie algebras produce solutions to the classical reflection equation.

Reflections on relative Rota-Baxter Lie algebras also yield solutions for certain Lie bialgebras.

Poisson Lie groups and homogeneous spaces are derived from these algebraic frameworks.

Abstract

In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $\lambda$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie bialgebra ($\lambda=0$) and factorizable Lie bialgebra ($\lambda\neq0$). Then we study reflections on relative Rota-Baxter Lie algebras, and also show that they give rise to solutions of the classical reflection equation for certain Lie bialgebras determined by the relative Rota-Baxter operators. In particular, involutive automorphisms on pre-Lie algebras and post-Lie algebras naturally lead to reflections on the induced relative Rota-Baxter Lie algebras. Finally, we derive Poisson Lie groups and Poisson homogeneous spaces from quadratic Rota-Baxter Lie algebras and relative Rota-Baxter Lie algebras.