Convergent Star Products on Cotangent Bundles of Lie Groups

TL;DR

This paper constructs a convergent star product on the cotangent bundle of a Lie group, extending it to a nuclear Fréchet algebra of analytic functions with holomorphic dependence on the deformation parameter.

Contribution

It introduces a nuclear Fréchet algebra framework for analytic functions on cotangent bundles of Lie groups, ensuring the continuity and holomorphic dependence of star products.

Findings

Star product converges on polynomial functions due to homogeneity.

Defined a nuclear Fréchet algebra of analytic functions on $T^*G$.

Proved the Gutt star product is continuous with holomorphic dependence on $ar$.

Abstract

For a connected real Lie group $G$ we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of $G$. This star product trivially converges on polynomial functions on $T^*G$ thanks to its homogeneity. We define a nuclear Fr\'echet algebra of certain analytic functions on $T^*G$, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter $\hbar$. This nuclear Fr\'echet algebra is realized as the completed (projective) tensor product of a nuclear Fr\'echet algebra of entire functions on $G$ with an appropriate nuclear Fr\'echet algebra of functions on $\mathfrak{g}^*$. The passage to the Weyl-ordered star product, i.e. the Gutt star product on $T^*G$, is shown to be preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on $\hbar$.