## Universal enveloping e_n-algebra

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

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

List some important properties of the universal enveloping -algebra.

**1**. As a commutative coalgebra is freely generated by .

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

**3.** Let be the (super) Lie algebra of homotopy groups of a topological pointed -connected space with respect to the Whitehead product: . Then is the chain complex of and is the chain complex of the n-fold based loop space of . For a parallelizable compact -manifold the manifoldic chain complex is the chain complex of the space of maps from to that sends the border to the marked point. For a manifold without borders the tautological morphism is the map induced by the embedding of to as the subspace of maps that factors through a point.

**4.** Denote by the operad that manages homotopy Lie algebras with the bracket of degree . There is a morphism that is Koszul dual to the embedding . Thus for any -algebra the complex is a -algebra. Denote it by . The functor is left adjoint to . It follows that for any -algebra there is the *canonical morphism* of -algebras .

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