nLab
coexponential map

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

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

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

y 1y k1k! σΣ(k)y^ σ(1)y^ σ(n) 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,,y kLS(L)y_1,\ldots, y_k\in L\hookrightarrow S(L) are arbitrary and y^ 1,,y^ k\hat{y}_1,\ldots,\hat{y}_k are the same elements understood in LU(L)L\hookrightarrow U(L). The coexponential map is an isomorphism of coalgebras S(L)U(L)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)U(L)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)