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

such that e(y) n=e(y n) for every “linear” element yLS(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) are arbitrary and y^ 1,,y^ k are the same elements understood in LU(L). The coexponential map is an isomorphism of coalgebras S(L)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) sometimes called also generalized symmetrization maps.

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