Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911783964377088 |
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| author | Hong, Yanyong Bai, Chengming Guo, Li |
| author_facet | Hong, Yanyong Bai, Chengming Guo, Li |
| contents | Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family $(A,\circ_q)$, for scalars $q$, of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of $(A,\circ_0)$ recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$. The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, $\mathcal{O}$-operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras Hong, Yanyong Bai, Chengming Guo, Li Quantum Algebra Mathematical Physics Representation Theory 16T10, 16T25, 16S80, 17B38, 13N10, 16W99, 17A30, 17D25 Generalizing S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra, a deformation family $(A,\circ_q)$, for scalars $q$, of Novikov algebras is constructed from what we call an admissible commutative differential algebra, by adding a second linear operator to the commutative differential algebra with certain admissibility condition. The case of $(A,\circ_0)$ recovers the construction of S. Gelfand. This admissibility condition also ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. This way, a deformation family of Novikov bialgebras is obtained, under the further condition that the two operators are bialgebra derivations. As a special case, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$. The close relations of the classical bialgebra theories with Manin triples, classical Yang-Baxter type equations, $\mathcal{O}$-operators, and pre-structures are expanded to the two new bialgebra theories, in a way that is compatible with the just established connection between the two bialgebras. As an application, Novikov bialgebras are obtained from admissible differential Zinbiel algebras. |
| title | Deformation families of Novikov bialgebras via differential antisymmetric infinitesimal bialgebras |
| topic | Quantum Algebra Mathematical Physics Representation Theory 16T10, 16T25, 16S80, 17B38, 13N10, 16W99, 17A30, 17D25 |
| url | https://arxiv.org/abs/2402.16155 |