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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.20276 |
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| _version_ | 1866913912052514816 |
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| author | Lang, Honglei Sheng, Yunhe |
| author_facet | Lang, Honglei Sheng, Yunhe |
| contents | In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $λ$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie bialgebra ($λ=0$) and factorizable Lie bialgebra ($λ\neq0$). 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces Lang, Honglei Sheng, Yunhe Mathematical Physics 17B38, 17B62, 53D17 In this paper, first we introduce the notion of reflections on quadratic Rota-Baxter Lie algebras of weight $λ$, and show that they give rise to solutions of the classical reflection equation for the corresponding triangular Lie bialgebra ($λ=0$) and factorizable Lie bialgebra ($λ\neq0$). 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. |
| title | Reflections on Rota-Baxter Lie algebras, the classical reflection equation and Poisson homogeneous spaces |
| topic | Mathematical Physics 17B38, 17B62, 53D17 |
| url | https://arxiv.org/abs/2506.20276 |