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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Contact reduction and groupoid actions
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by Marco Zambon and Chenchang Zhu PDF
Trans. Amer. Math. Soc. 358 (2006), 1365-1401 Request permission

Abstract:

We introduce a new method to perform reduction of contact manifolds that extends Willett’s and Albert’s results. To carry out our reduction procedure all we need is a complete Jacobi map $J:M \rightarrow \Gamma _0$ from a contact manifold to a Jacobi manifold. This naturally generates the action of the contact groupoid of $\Gamma _0$ on $M$, and we show that the quotients of fibers $J^{-1}(x)$ by suitable Lie subgroups $\Gamma _x$ are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett’s reduced spaces are prequantizations of our reduced spaces; hence the former are completely determined by the latter. Since a symplectic manifold is prequantizable iff the symplectic form is integral, this explains why Willett’s reduction can be performed only at distinguished points. As an application we obtain Kostant’s prequantizations of coadjoint orbits. Finally we present several examples where we obtain classical contact manifolds as reduced spaces.
References
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Additional Information
  • Marco Zambon
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
  • Address at time of publication: Institut für Mathematik, Universität Zürich-Irchel, 8057 Zürich, Switzerland
  • Email: zambon@math.unizh.ch
  • Chenchang Zhu
  • Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
  • Address at time of publication: D-MATH, ETH-Zentrum, CH-8092 Zürich, Switzerland
  • Email: zhu@math.ethz.ch
  • Received by editor(s): May 25, 2004
  • Published electronically: June 21, 2005
  • © Copyright 2005 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 1365-1401
  • MSC (2000): Primary 53D10, 53D20, 58H05
  • DOI: https://doi.org/10.1090/S0002-9947-05-03832-8
  • MathSciNet review: 2187657