Poincaré-Birkhoff-Witt Theorems in Higher Algebra

Abstract

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.

To appear in the Philosophical Transactions of the Royal Society A. 16 pages