Nikita Markarian’s mathblog

Universal enveloping e_n-algebra

Posted in Mathematics by nikitamarkarian on July 30, 2011

The universal enveloping e_n-algebra is a functor from the category of homotopy Lie algebras over Q to the category of algebras over the operad of rational chains of the operad of little n-discs. For n=1 one get (the derived functor of) the usual universal enveloping algebra.

Given a homotopy Lie algebra l, the Quillen functor produces from it a homotopy cocomutative algebra, that is a comodule over the operad e_\infty. This coalgebra is the homological Chevalley complex of l, denote it by Ch_*(l). Due to the embedding e_n\to e_\infty one may treat Ch_*(l) as a co-e_n-algebra and further as a co-e_n-algebra in the category of homotopy cocommutative algebras. Applying the Koszul duality for e_n-algebras to this co-e_n-algebra we get a e_n-algebra in the category of homotopy cocommutative algebras. In other words, we get a e_n-algebra with compatible homotopy cocommutative coalgebra structure. Call it the universal enveloping e_n-algebra of l and denote it by U^n(l).

List some important properties of the universal enveloping e_n-algebra.

1. As a commutative coalgebra U^n(l) is freely generated by l[n-1].

2. For a compact parallelized n-manifold without borders M consider the manifoldic chain complex CM_*(U^n(l)). By the Koszul-Poincare duality the latter complex is isomorphic to the manifoldic chain complex of Ch_*(l) being considered as a co-e_n-algebra (one may define manifoldic homology of a co-e_n-algebra in same way as for an e_n-algebra). There is a tautological morphism from Ch_*(l) to the latter complex. Thus we get a morphism Ch_*(l)\to CM_*(U^n(l)), call it the tautological morphism. If n=1 and manifold M is a circle for a Lie algebra l the tautological morphism is the morphism from the Chevalley chain complex of l to the Hochschild chain complex of the universal enveloping algebra.

3. Let l be the (super) Lie algebra of homotopy groups of a topological pointed n-connected space X with respect to the Whitehead product: l_i=\pi_{1-i}(X). Then Ch_*(l) is the chain complex of X and U^n(l) is the chain complex of the n-fold based loop space of X. For a parallelizable compact n-manifold M the manifoldic chain complex CM_*(U^n(l)) is the chain complex of the space of maps from M to X that sends the border to the marked point. For a manifold without borders the tautological morphism is the map C_*(X)\to CM_*(U^n(l))=C_*(Map (M,X)) induced by the embedding of X to Map(M,X) as the subspace of maps that factors through a point.

4. Denote by L_\infty[n] the operad that manages homotopy Lie algebras with the bracket of degree n. There is a morphism L_\infty[1-n]\to e_n that is Koszul dual to the embedding e_n\to e_\infty. Thus for any e_n-algebra A the complex A[n-1] is a L_\infty-algebra. Denote it by L(A). The functor U^n(-) is left adjoint to L(-). It follows that for any e_n-algebra there is the canonical morphism of e_n-algebras U^n(L(A))\to A.

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