Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913530039500800 |
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| author | Lin, Yuanchang Zhou, Peng Bai, Chengming |
| author_facet | Lin, Yuanchang Zhou, Peng Bai, Chengming |
| contents | It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13230 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras Lin, Yuanchang Zhou, Peng Bai, Chengming Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 16T10, 16T25, 16W99, 17A30, 17A60, 17B38, 17B60, 17B62, 17B65, 17D25 It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given. |
| title | Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 16T10, 16T25, 16W99, 17A30, 17A60, 17B38, 17B60, 17B62, 17B65, 17D25 |
| url | https://arxiv.org/abs/2409.13230 |