On differential Hopf algebras and $B_\infty$ algebras
| Autor: | Gálvez-Carrillo, Imma, Ronco, María, Tonks, Andy |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 18pp |
| Databáze: | arXiv |
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