Geometry & Topology

Higher enveloping algebras

Ben Knudsen

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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.

Article information

Geom. Topol., Volume 22, Number 7 (2018), 4013-4066.

Received: 2 March 2017
Revised: 12 March 2018
Accepted: 23 April 2018
First available in Project Euclid: 14 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17B99: None of the above, but in this section 55R80: Discriminantal varieties, configuration spaces 55P35: Loop spaces

Lie algebra enveloping algebra configuration space factorization homology


Knudsen, Ben. Higher enveloping algebras. Geom. Topol. 22 (2018), no. 7, 4013--4066. doi:10.2140/gt.2018.22.4013.

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