# nLab coexponential map

The coexponential or symmetrization map is the unique linear map from the symmetric Hopf algebra $S\left(L\right)$ on the (underlying vector space of a) Lie algebra $L$ to the enveloping Hopf algebra $U\left(L\right)$ of $L$

$e:S\left(L\right)\to U\left(L\right)$e : S(L)\to U(L)

such that $e\left(y{\right)}^{n}=e\left({y}^{n}\right)$ for every “linear” element $y\in L↪S\left(L\right)$. This map is also given by the formula

${y}_{1}\cdots {y}_{k}↦\frac{1}{k!}\sum _{\sigma \in \Sigma \left(k\right)}{\stackrel{^}{y}}_{\sigma \left(1\right)}\cdots {\stackrel{^}{y}}_{\sigma \left(n\right)}$y_1\cdots y_k \mapsto \frac{1}{k!} \sum_{\sigma\in\Sigma(k)}\hat{y}_{\sigma(1)}\cdots \hat{y}_{\sigma(n)}

where ${y}_{1},\dots ,{y}_{k}\in L↪S\left(L\right)$ are arbitrary and ${\stackrel{^}{y}}_{1},\dots ,{\stackrel{^}{y}}_{k}$ are the same elements understood in $L↪U\left(L\right)$. The coexponential map is an isomorphism of coalgebras $S\left(L\right)\cong U\left(L\right)$.

The term symmetrization map is more widely used but it is a bit imprecise as it is used also for more general coalgebra isomorphisms $S\left(L\right)\to U\left(L\right)$ sometimes called also generalized symmetrization maps.

Revised on May 16, 2013 18:38:10 by Zoran Škoda (161.53.130.104)