Poincar\'{e}-Birkhoff-Witt Theorems in Higher Algebra

TL;DR

This paper generalizes the classical Poincaré-Birkhoff-Witt theorem to the setting of spectral Lie algebras and higher algebra, establishing new relations between operads in spectra and providing constructions for higher enveloping algebras.

Contribution

It introduces a higher algebra version of the PBW theorem for spectral Lie algebras and explores fundamental relations between different $ ext{E}_n$-operads.

Findings

Established a PBW-type theorem for spectral Lie algebras.

Derived relations between operads in spectra, including the commutative and associative operads.

Provided a construction for higher enveloping $ ext{E}_n$-algebras.

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.