1. Poincar\'{e}-Birkhoff-Witt Theorems in Higher Algebra
- Author
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Antolín-Camarena, Omar, Brantner, Lukas, and Heuts, Gijs
- Subjects
Mathematics - Algebraic Topology ,Mathematics - Category Theory ,Mathematics - Representation Theory ,17B01, 17B05, 17B35, 18N70 - Abstract
We extend the classical Poincar\'e-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 Poincar\'{e}--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., Comment: 14 pages
- Published
- 2025