Exponentiations over the universal enveloping algebra of \(\text{sl}_2(\mathbb C)\)
type: Item
instance of: scholarly article
title: Exponentiations over the universal enveloping algebra of \(\text{sl}_2(\mathbb C)\)
zbMATH Open document ID: 1232.17023
author: Sonia L'Innocente
author: Françoise Point
author: Angus Macintyre
published in: Annals of Pure and Applied Logic
publication date: 2011-08-26T00:00:00Z
review text: 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 met…
reviewed by: Carlo Toffalori
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 5943667
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: exponential map
zbMATH Keywords: ultraproduct
zbMATH Keywords: asymptotic cone
MaRDI profile type: Publication
full work available at URL: j.apal.2010.06.006
OpenAlex ID: W2008421959
cites work: Q2761411
cites work: Model theory
cites work: Dimension of asymptotic cones of Lie groups
cites work: Primitive ideals in enveloping algebras
cites work: Q5472809
cites work: Q5666543
cites work: Pseudo-finite dimensional representations of \(\text{sl}(2,k)\).
cites work: The nonstandard quantum plane
cites work: Q4303969
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Asymptotic Cones and Ultrapowers of Lie Groups
cites work: Q3509170
cites work: RINGS OF DEFINABLE SCALARS OF VERMA MODULES
cites work: Model theory of exponentials on Lie algebras
cites work: Lie groups. An introduction through linear groups
cites work: Lie algebras and Lie groups. 1964 lectures, given at Harvard University.
cites work: Exponential rings, exponential polynomials and exponential functions
DOI: 10.1016/J.APAL.2010.06.006
description: scientific article; zbMATH DE number 5943667