nLab 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 $y_1,\ldots, y_k\in L\hookrightarrow S(L)$ are arbitrary and $\hat{y}_1,\ldots,\hat{y}_k$ 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.