Abstract
We review some of the main achievements of the orbit method, when applied to Poisson–Lie groups and Poisson homogeneous spaces or spaces with an invariant Poisson structure. We consider C∗-algebra quantization obtained through groupoid techniques, and we try to put the results obtained in algebraic or representation theoretical contexts in relation with groupoid quantization.
Highlights
Our Poisson manifolds will not be arbitrary ones; we rather concentrate on examples characterized by the presence of a wide array of symmetries, describing some ongoing work and various open questions
We review the easiest cases, just to give some comprehension of the set of results needed to verify in which cases the quantum orbit method holds true
Let us consider the case of a linear Poisson structure, i.e., the usual Poisson structure on the linear dual of a Lie algebra g∗ where Lie brackets are given in terms of structural constant of g as: n
Summary
Kirillov [1], the recipes the method provides are far from accurate, sometimes wrong or need corrections and modifications, and are usually difficult to transform into rigorous proofs. Despite these drawbacks, it provides an illuminating guide, or, to cite another of the heroes of its study, David Vogan, a “spoiled treasure map”, parts of which are pretty well understood, while others point towards unknown roads [2]. Our Poisson manifolds will not be arbitrary ones; we rather concentrate on examples characterized by the presence of a wide array of symmetries, describing some ongoing work and various open questions
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