Higher enveloping algebras

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 Poincaré–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.

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Keywords

Lie algebra, enveloping algebra, configuration space, factorization homology

Mathematical Subject Classification 2010

Primary: 17B99, 55R80, 55P35

References

Publication

Received: 2 March 2017

Revised: 12 March 2018

Accepted: 23 April 2018

Published: 6 December 2018

Proposed: Haynes R Miller

Seconded: Ulrike Tillmann, Peter Teichner

Authors

Ben Knudsen
Department of Mathematics Harvard University Cambridge, MA United States

Volume 22, issue 7 (2018)

Ben Knudsen

Abstract