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

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arxiv 2506.20276 v1 pith:SOKTZLAM submitted 2025-06-25 math-ph math.MP

Honglei Lang , Yunhe Sheng This is my paper
classification math-phmath.MP
keywords algebrasrota-baxterreflectionsrelativeclassicalequationlambdapoisson
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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.

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