Higher enveloping algebras

TL;DR

This paper develops a theory of higher enveloping algebras for spectral Lie algebras over operads of little G-framed disks, generalizing classical results and linking to configuration space geometry.

Contribution

It introduces a universal characterization and a PBW-type formula for higher enveloping algebras, extending Lie theory to higher algebraic structures with geometric applications.

Findings

Higher enveloping algebras characterized by a universal property.

A generalized PBW theorem for these algebras.

Stable homotopy types of configuration spaces are proper homotopy invariants.

Abstract

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincar\'{e}-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.