Journal of Lie Theory 27 (2017), No. 2, 579--622
Copyright Heldermann Verlag 2017
Convergence of the Gutt Star Product
Chiara Esposito
Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, 97074 Würzburg, Germany
chiara.esposito@mathematik.uni-wuerzburg.de
Paul Stapor
Helmholtz Zentrum München, Deutsches Forschungszentrum für Gesundheit und Umwelt, Institute of Computational Biology, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany
paul.stapor@helmholtz-muenchen.de
Stefan Waldmann
Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, 97074 Würzburg, Germany
stefan.waldmann@mathematik.uni-wuerzburg.de
We consider the Gutt star product viewed as an associative deformation of the symmetric algebra SA(g) over a Lie algebra g and discuss its continuity properties: we establish a locally convex topology on SA(g) such that the Gutt star product becomes continuous. Here we have to assume a mild technical condition on g: it has to be an Asymptotic Estimate Lie algebra. This condition is e.g. fulfilled automatically for all finite-dimensional Lie algebras. The resulting completion of the symmetric algebra can be described explicitly and yields not only a locally convex algebra but also the Hopf algebra structure maps inherited from the universal enveloping algebra are continuous. We show that all Hopf algebra structure maps depend analytically on the deformation parameter. The construction enjoys good functorial properties.
Keywords: Gutt star product, convergence, locally convex algebras, universal enveloping algebra.
MSC: 53D55, 46H05, 46A03, 16S30.
[ Fulltext-pdf (523 KB)] for subscribers only.