Convergence of the Gutt Star Product

Chiara Esposito¹ ; Paul Stapor² ; Stefan Waldmann¹

¹Institut für Mathematik, Universität Würzburg, Campus Hubland Nord, Emil-Fischer-Str. 31, 97074 Würzburg, Germany
²Helmholtz Zentrum München, Deutsches Forschungszentrum für Gesundheit und Umwelt, Institute of Computational Biology, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany

Journal of Lie Theory, Tome 27 (2017) no. 2, pp. 579-622

Résumé

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.

Détail

Classification : 53D55, 46H05, 46A03, 16S30
Mots-clés : Gutt star product, convergence, locally convex algebras, universal enveloping algebra