We provide spectral Lie algebras with enveloping algebras over the operad of little
–framed
–dimensional disks for
any choice of dimension
and structure group ,
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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