coexponential map

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

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

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

$${y}_{1}\cdots {y}_{k}\mapsto \frac{1}{k!}\sum _{\sigma \in \Sigma (k)}{\hat{y}}_{\sigma (1)}\cdots {\hat{y}}_{\sigma (n)}$$

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

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

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