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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.03116 |
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| _version_ | 1866913989809668096 |
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| author | Antolín-Camarena, Omar Brantner, Lukas Heuts, Gijs |
| author_facet | Antolín-Camarena, Omar Brantner, Lukas Heuts, Gijs |
| contents | We extend the classical Poincaré-Birkhoff-Witt theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is the quotient of the associative operad by a right action of the spectral Lie operad. This statement, in turn, is a consequence of a fundamental relation between different $\mathbb{E}_n$-operads, which we articulate and prove. We deduce a variant of the Poincaré--Birkhoff--Witt theorem for relative enveloping algebras of $\mathbb{E}_n$-algebras. Our methods also give a simple construction and description of the higher enveloping $\mathbb{E}_n$-algebras of a spectral Lie algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Poincaré-Birkhoff-Witt Theorems in Higher Algebra Antolín-Camarena, Omar Brantner, Lukas Heuts, Gijs Algebraic Topology Category Theory Representation Theory 17B01, 17B05, 17B35, 18N70 We extend the classical Poincaré-Birkhoff-Witt theorem to higher algebra by establishing a version that applies to spectral Lie algebras. We deduce this statement from a basic relation between operads in spectra: the commutative operad is the quotient of the associative operad by a right action of the spectral Lie operad. This statement, in turn, is a consequence of a fundamental relation between different $\mathbb{E}_n$-operads, which we articulate and prove. We deduce a variant of the Poincaré--Birkhoff--Witt theorem for relative enveloping algebras of $\mathbb{E}_n$-algebras. Our methods also give a simple construction and description of the higher enveloping $\mathbb{E}_n$-algebras of a spectral Lie algebra. |
| title | Poincaré-Birkhoff-Witt Theorems in Higher Algebra |
| topic | Algebraic Topology Category Theory Representation Theory 17B01, 17B05, 17B35, 18N70 |
| url | https://arxiv.org/abs/2501.03116 |