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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Construction of canonical coordinates for exponential Lie groups


Authors: Didier Arnal, Bradley Currey and Bechir Dali
Journal: Trans. Amer. Math. Soc. 361 (2009), 6283-6348
MSC (2000): Primary 22E25, 22E27; Secondary 53D50
Published electronically: July 22, 2009
MathSciNet review: 2538595
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Abstract: Given an exponential Lie group $ G$, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Hölder basis. Thus we obtain a stratification of $ \mathfrak{g}^*$ into $ G$-invariant algebraic subsets, and for each such subset $ \Omega$, an explicit cross-section $ \Sigma \subset \Omega$ for coadjoint orbits in $ \Omega$, so that each pair $ (\Omega, \Sigma)$ behaves predictably under the associated restriction maps on $ \mathfrak{g}^*$. The cross-section mapping $ \sigma : \Omega \rightarrow \Sigma$ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with $ \ell \in \Omega$. For each $ \Omega$, algebras $ \mathcal E^0(\Omega)$ and $ \mathcal E^1(\Omega)$ of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way.

Now let $ 2d > 0$ be the dimension of coadjoint orbits in $ \Omega$. An explicit algorithm is given for the construction of complex-valued real analytic functions $ \{q_1,q_2, \dots , q_d\}$ and $ \{p_1, p_2, \dots, p_d\}$ such that on each coadjoint orbit $ \mathcal{O}$ in $ \Omega$, the canonical 2-form is given by $ \sum dp_k \wedge dq_k$. The functions $ \{q_1,q_2, \dots , q_d\}$ belong to $ \mathcal E^0(\Omega)$, and the functions $ \{p_1, p_2, \dots, p_d\}$ belong to $ \mathcal E^1(\Omega)$. The associated geometric polarization on each orbit $ \mathcal{O}$ coincides with the complex Vergne polarization, and a global Darboux chart on $ \mathcal{O}$ is obtained in a simple way from the coordinate functions $ (p_1, \dots, p_d,q_1, \dots , q_d)$ (restricted to $ \mathcal{O}$). Finally, the linear evaluation functions $ \ell \mapsto \ell(X)$ are shown to be quantizable as well.


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Additional Information

Didier Arnal
Affiliation: Institut de Mathématiques de Bourgogne, Université de Bourgogne, CNRS UMR 5584, BP47870, F-21078 Dijon Cedex, France
Email: didier.arnal@u-bourgogne.fr

Bradley Currey
Affiliation: Department of Mathematics and Computer Science, St. Louis University, St. Louis, Missouri 63103
Email: curreybn@slu.edu

Bechir Dali
Affiliation: Département de Mathématiques, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisie
Email: bechir.dali@fss.rnu.tn

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04936-8
PII: S 0002-9947(09)04936-8
Received by editor(s): July 2, 2007
Published electronically: July 22, 2009
Additional Notes: The first author would like to thank the CNRS for its support and the Faculté de Sciences de Bizerte for its hospitality.
The second author would like to thank the Université de Bourgogne for their hospitality and support.
The third author would like to thank the Université de Bourgogne for its hospitality during his stay. He would also like to thank the DGRSRT of Tunisia for its support.
Article copyright: © Copyright 2009 American Mathematical Society