A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras
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| Format: | Preprint |
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2024
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| _version_ | 1866911763935526912 |
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| author | Hong, Yangyon Bai, Chengming Guo, Li |
| author_facet | Hong, Yangyon Bai, Chengming Guo, Li |
| contents | Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of $\mathcal{O}$-operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13608 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras Hong, Yangyon Bai, Chengming Guo, Li Quantum Algebra Mathematical Physics 17A30, 17B62, 17B69, 17B38, 17D25, 81T40 Gel'fand-Dorfman algebras (GD algebras) give a natural construction of Lie conformal algebras and are in turn characterized by this construction. In this paper, we define the Gel'fand-Dorfman bialgebra (GD bialgebras) and enrich the above construction to a construction of Lie conformal bialgebras by GD bialgebras. As a special case, Novikov bialgebras yield Lie conformal bialgebras. We further introduce the notion of the Gel'fand-Dorfman Yang-Baxter equation (GDYBE), whose skew-symmetric solutions produce GD bialgebras. Moreover, the notions of $\mathcal{O}$-operators on GD algebras and pre-Gel'fand-Dorfman algebras (pre-GD algebras) are introduced to provide skew-symmetric solutions of the GDYBE. The relationships between these notions for GD algebras and the corresponding ones for Lie conformal algebras are given. In particular, there is a natural construction of Lie conformal bialgebras from pre-GD algebras. Finally, GD bialgebras are characterized by certain matched pairs and Manin triples of GD algebras. |
| title | A bialgebra theory of Gel'fand-Dorfman algebras with applications to Lie conformal bialgebras |
| topic | Quantum Algebra Mathematical Physics 17A30, 17B62, 17B69, 17B38, 17D25, 81T40 |
| url | https://arxiv.org/abs/2401.13608 |