# 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 $U^n(L(A))\to A$.