Convergent Star Products on Cotangent Bundles of Lie Groups

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 Fréchet 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 . This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on 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 .

Strict quantization of polynomial Poisson structures

“…In this case, convergence on polynomials is immediate, since for all f, g ∈ P(R d ) their formal star product f ⋆ g is a polynomial in the formal parameter which can be evaluated to any value of . The continuity of star products, and the properties of the algebra obtained by completion, was studied successfully for constant and linear Poisson structures [25,10], also in infinite-dimensional, field-theoretic [23] and "global" settings, such as on coadjoint orbits of Lie groups [17,21] or on cotangent bundles of Lie groups [13] (see [26] for a review). Yet, although constant and linear Poisson structures are important classes of Poisson structures, one cannot expect them to cover all physically relevant phase spaces.…”