Higher Enveloping Algebras and Configuration Spaces of Manifolds

We provide Lie algebras with enveloping algebras over the operad of little n-dimensional disks for any choice of n, and we give two complementary descriptions of these objects. 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, while the second is a result analogous to the classical Poincare-Birkhoff-Witt theorem, giving a concrete identification of this algebra in terms of Lie algebra homology.;We apply this theory to the study of configuration spaces. Using the technique of factorization homology, we realize the rational homology of the configuration spaces of an arbitrary manifold M as the homology of a Lie algebra constructed from the compactly supported cohomology of M. Consideration of the Chevalley-Eilenberg complex of this Lie algebra leads to extensions of theorems of Boedigheimer-Cohen-Taylor and Felix-Thomas, to a new, combinatorial proof of the homological stability results of Church and Randal-Williams, and to explicit computations.