A bialgebra theory of post-Lie algebras via Manin triples and generalized Hessian Lie groups
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2025
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| author | Lu, Dilei Bai, Chengming Guo, Li |
| author_facet | Lu, Dilei Bai, Chengming Guo, Li |
| contents | We develop a bialgebra theory of post-Lie algebras that can be characterized by Manin triples of post-Lie algebras associated to a bilinear form satisfying certain invariant conditions. In the absence of dual representations for adjoint representations of post-Lie algebras, we utilize the geometric interpretation of post-Lie algebras to find the desired invariant condition, by generalizing pseudo-Hessian Lie groups to allow constant torsion for the flat connection. The resulting notion is a generalized pseudo-Hessian post-Lie algebra, which is a post-Lie algebra equipped with a nondegenerate symmetric invariant bilinear form. Moreover, generalized pseudo-Hessian post-Lie algebras are also naturally obtained from quadratic Rota-Baxter Lie algebras of weight one. On the other hand, the notion of partial-pre-post-Lie algebra (pp-post-Lie algebras) is introduced as the algebraic structure underlying generalized pseudo-Hessian post-Lie algebras, by splitting one of the two binary operations of post-Lie algebras. The notion of pp-post-Lie bialgebras is introduced as the equivalent structure of Manin triples of post-Lie algebras associated to a nondegenerate symmetric invariant bilinear form, thereby establishing a bialgebra theory for post-Lie algebras via the Manin triple approach. We also study the related analogs of the classical Yang-Baxter equation, $\mathcal O$-operators and successors for pp-post-Lie algebras. In particular, there is a construction of pp-post-Lie bialgebras from the successors of pp-post-Lie algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_04954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A bialgebra theory of post-Lie algebras via Manin triples and generalized Hessian Lie groups Lu, Dilei Bai, Chengming Guo, Li Quantum Algebra Differential Geometry Rings and Algebras 17D25, 17B62, 22E60, 17B38, 58D17, 53C05, 16T10, 17A30, 17A36 We develop a bialgebra theory of post-Lie algebras that can be characterized by Manin triples of post-Lie algebras associated to a bilinear form satisfying certain invariant conditions. In the absence of dual representations for adjoint representations of post-Lie algebras, we utilize the geometric interpretation of post-Lie algebras to find the desired invariant condition, by generalizing pseudo-Hessian Lie groups to allow constant torsion for the flat connection. The resulting notion is a generalized pseudo-Hessian post-Lie algebra, which is a post-Lie algebra equipped with a nondegenerate symmetric invariant bilinear form. Moreover, generalized pseudo-Hessian post-Lie algebras are also naturally obtained from quadratic Rota-Baxter Lie algebras of weight one. On the other hand, the notion of partial-pre-post-Lie algebra (pp-post-Lie algebras) is introduced as the algebraic structure underlying generalized pseudo-Hessian post-Lie algebras, by splitting one of the two binary operations of post-Lie algebras. The notion of pp-post-Lie bialgebras is introduced as the equivalent structure of Manin triples of post-Lie algebras associated to a nondegenerate symmetric invariant bilinear form, thereby establishing a bialgebra theory for post-Lie algebras via the Manin triple approach. We also study the related analogs of the classical Yang-Baxter equation, $\mathcal O$-operators and successors for pp-post-Lie algebras. In particular, there is a construction of pp-post-Lie bialgebras from the successors of pp-post-Lie algebras. |
| title | A bialgebra theory of post-Lie algebras via Manin triples and generalized Hessian Lie groups |
| topic | Quantum Algebra Differential Geometry Rings and Algebras 17D25, 17B62, 22E60, 17B38, 58D17, 53C05, 16T10, 17A30, 17A36 |
| url | https://arxiv.org/abs/2502.04954 |