Convergence of the Gutt star product. (English) Zbl 1375.53110

Esposito, Chiara; Stapor, Paul; Waldmann, Stefan

J. Lie Theory 27, No. 2, 579-622 (2017).

Authors’ abstract: We consider the Gutt star product viewed as an associative deformation of the symmetric algebra \(S^{\ast}({\mathfrak g})\) over a Lie algebra \({\mathfrak g}\) and discuss its continuity properties: we establish a locally convex topology on \(S^{\ast}({\mathfrak g})\) such that the Gutt star product becomes continuous. Here we have to assume a mild technical condition on \({\mathfrak g}\): it has to be an asymptotic estimate Lie algebra. This condition is fulfilled automatically, e.g., 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.

Reviewer: Benjamin Cahen (Metz)

MSC:

53D55	Deformation quantization, star products
46H05	General theory of topological algebras
46A03	General theory of locally convex spaces
16S30	Universal enveloping algebras of Lie algebras

Keywords:

Gutt star-product; convergence; locally convex algebras; universal enveloping algebra