2025/01/06 by Antolín-Camarena, Omar, Lukas Brantner, Brantner, Lukas +2
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic structures and combinatorial models #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2501.03116
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 𝔼n-operads, which we articulate and prove. We deduce a variant of the Poincaré--Birkhoff--Witt theorem for relative enveloping algebras of 𝔼n-algebras. Our methods also give a simple construction and description of the higher enveloping 𝔼n-algebras of a spectral Lie algebra.