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

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.

### 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
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