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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2409.06632 |
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| _version_ | 1866917772427001856 |
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| author | Gálvez-Carrillo, Imma Ronco, María Tonks, Andy |
| author_facet | Gálvez-Carrillo, Imma Ronco, María Tonks, Andy |
| contents | We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_\infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_\infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_\infty$ structures are compatible, and provide an appropriate $B_\infty$ structure for the structure theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On differential Hopf algebras and $B_\infty$ algebras Gálvez-Carrillo, Imma Ronco, María Tonks, Andy Quantum Algebra 16T05, 18M70 (Primary) 17A30, 55S20, 81R60 (Secondary) We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure that restricts to the subspace of primitives, and conversely $H$ may be recovered via a universal enveloping differential-2-associative algebra. This extends the work of Loday and Ronco [12] where the ungraded non-differential case was treated, and only the multibrace part of the $B_\infty$ structure was found. We show that the multibrace structure of [12] originates from a twisting of a quasi-trivial structure, extending the work of Markl [14] on the $A_\infty$ structure underlying any algebra with a square-zero endomorphism. In this framework it is also clear that the multibrace and $A_\infty$ structures are compatible, and provide an appropriate $B_\infty$ structure for the structure theorem. |
| title | On differential Hopf algebras and $B_\infty$ algebras |
| topic | Quantum Algebra 16T05, 18M70 (Primary) 17A30, 55S20, 81R60 (Secondary) |
| url | https://arxiv.org/abs/2409.06632 |