Exponentiations over the universal enveloping algebra of \(\text{sl}_2(\mathbb C)\)
Let \(\text{sl}_2(\mathbb C)\) be the Lie algebra of \(2 \times 2\) traceless matrices with entries in the complex field \(\mathbb C\) and let \(U\) denote its universal enveloping algebra. Recall that an exponential map \(exp\) can be defined on the matrix rings \(M_{\lambda + 1}({\mathbb C})\) (with \(\lambda\) a non-negative integer) in terms of power series, taking any matrix \(A\) into \(\exp(A)= \sum_{k=0}^{ \infty} {{A^k} \over {k!}}\). Using this and the universal property of \(U\), the authors introduce a sequence of exponential maps indexed by \(\lambda\) from \(U\) to \(\text{GL}_{\lambda + 1}(\mathbb C)\). Various properties of these maps are studied, and elements in their kernels and images are explicitly calculated. Then it is proved that, for every non-principal ultrafilter \(F\) on \(\omega\), \(U\) embeds into the corresponding ultraproduct \(\prod_F M_{\lambda + 1} ({\mathbb C})\), which leads to define a new natural exponential map \(\text{EXP}\) from \(U\) to \(\prod_F \text{GL}_{\lambda +1} (\mathbb C)\). It is shown that this equips \(U\) with the structure of a (non-commutative) exponential ring. Moreover part of the kernel of \(EXP\) is again explicitly calculated. The function \(\text{EXP}\) is also compared with another exponential map, defined by Serre in the completion \(\hat{U}\) of \(U\) on the ideal of \(\hat{U}\) generated by the (standard) generators of \(U\). A suitable norm in \(\prod_F M_{\lambda +1} (\mathbb C)\) (taking its values in a non-standard ultrapower of the real field) endows \(U\) with a natural topology, which, together with the corresponding topology on the ultraproduct of the \(\text{GL}_{\lambda + 1}(\mathbb C)\), makes \(\text{EXP}\) a continuous map and the subgroup generated by \(\text{EXP}(U)\) a topological group. Finally, by using another norm on the various \(M_{\lambda + 1}(\mathbb C)\), the asymptotic cone relative to that norm and again non-principal ultrafilters, the authors embed \(U\) in a complete metric space, on which \(U\) has a faithful continuous action.
scientific article; zbMATH DE number 5943667