Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras
We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras, by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double construc...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211866 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras. Yuanchang Lin, Xuguang Liu and Chengming Bai. SIGMA 19 (2023), 018, 47 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862588454000394240 |
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| author | Lin, Yuanchang Liu, Xuguang Bai, Chengming |
| author_facet | Lin, Yuanchang Liu, Xuguang Bai, Chengming |
| citation_txt | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras. Yuanchang Lin, Xuguang Liu and Chengming Bai. SIGMA 19 (2023), 018, 47 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras, by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of the associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of -operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras, and the set of coherent derivations on an ASI bialgebra composes a Lie algebra, which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well-constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras.
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| first_indexed | 2026-03-13T19:01:36Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211866 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T19:01:36Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
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| spelling | Lin, Yuanchang Liu, Xuguang Bai, Chengming 2026-01-14T09:52:57Z 2023 Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras. Yuanchang Lin, Xuguang Liu and Chengming Bai. SIGMA 19 (2023), 018, 47 pages 1815-0659 2020 Mathematics Subject Classification: 16T10; 16T25; 16W99; 17A30; 17B62; 57R56; 81R60 arXiv:2207.00390 https://nasplib.isofts.kiev.ua/handle/123456789/211866 https://doi.org/10.3842/SIGMA.2023.018 We establish a bialgebra theory for differential algebras, called differential antisymmetric infinitesimal (ASI) bialgebras, by generalizing the study of ASI bialgebras to the context of differential algebras, in which the derivations play an important role. They are characterized by double constructions of differential Frobenius algebras as well as matched pairs of differential algebras. Antisymmetric solutions of an analogue of the associative Yang-Baxter equation in differential algebras provide differential ASI bialgebras, whereas in turn the notions of -operators of differential algebras and differential dendriform algebras are also introduced to produce the former. On the other hand, the notion of a coherent derivation on an ASI bialgebra is introduced as an equivalent structure of a differential ASI bialgebra. They include derivations on ASI bialgebras, and the set of coherent derivations on an ASI bialgebra composes a Lie algebra, which is the Lie algebra of the Lie group consisting of coherent automorphisms on this ASI bialgebra. Finally, we apply the study of differential ASI bialgebras to Poisson bialgebras, extending the construction of Poisson algebras from commutative differential algebras with two commuting derivations to the context of bialgebras, which is consistent with the well-constructed theory of Poisson bialgebras. In particular, we construct Poisson bialgebras from differential Zinbiel algebras. This work is supported by NSFC (11931009, 12271265, 12261131498), the Fundamental Research Funds for the Central Universities, and Nankai Zhide Foundation. The authors thank the referees for their valuable suggestions to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras Article published earlier |
| spellingShingle | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras Lin, Yuanchang Liu, Xuguang Bai, Chengming |
| title | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras |
| title_full | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras |
| title_fullStr | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras |
| title_full_unstemmed | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras |
| title_short | Differential Antisymmetric Infinitesimal Bialgebras, Coherent Derivations and Poisson Bialgebras |
| title_sort | differential antisymmetric infinitesimal bialgebras, coherent derivations and poisson bialgebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211866 |
| work_keys_str_mv | AT linyuanchang differentialantisymmetricinfinitesimalbialgebrascoherentderivationsandpoissonbialgebras AT liuxuguang differentialantisymmetricinfinitesimalbialgebrascoherentderivationsandpoissonbialgebras AT baichengming differentialantisymmetricinfinitesimalbialgebrascoherentderivationsandpoissonbialgebras |