By Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)

ISBN-10: 1461217709

ISBN-13: 9781461217701

ISBN-10: 1461272742

ISBN-13: 9781461272748

This booklet is an outgrowth of the actions of the heart for Geometry and Mathematical Physics (CGMP) at Penn country from 1996 to 1998. the guts used to be created within the arithmetic division at Penn country within the fall of 1996 for the aim of selling and aiding the actions of researchers and scholars in and round geometry and physics on the collage. The CGMP brings many viewers to Penn kingdom and has ties with different study teams; it organizes weekly seminars in addition to annual workshops The booklet includes 17 contributed articles on present study subject matters in a number of fields: symplectic geometry, quantization, quantum teams, algebraic geometry, algebraic teams and invariant conception, and personality istic sessions. many of the 20 authors have talked at Penn country approximately their study. Their articles current new effects or speak about fascinating perspec tives on contemporary paintings. the entire articles were refereed within the standard model of fine clinical journals. Symplectic geometry, quantization and quantum teams is one major subject matter of the booklet. a number of authors examine deformation quantization. As tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting transverse polarizations, and reviews the instant map on the subject of semisimple coadjoint orbits. Bieliavsky constructs an particular star-product on holonomy reducible sym metric coadjoint orbits of an easy Lie crew, and he indicates how one can con struct a star-representation which has fascinating holomorphic properties.

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Ucdavis . edu Ranee Brylinski Department of Mathematics Penn State University E-mail address: rkblDmath. psu. edu Received 11/23/97; revised 8/5/97 THE GEOMETRY SURROUNDING THE ARNOLD-LIOUVILLE THEOREM AUGUSTIN BANYAGA ABSTRACT. e. a fibration 7r : M2n -+ wn of a symplectic 2n-dimensional manifold M over an n manifold W with isotropic tori of various dimensions as fibers. This definition , which contains as particular cases, completely integrable hamiltonian systems, hamiltonian actions, and Duistermaat (lagrangian) fibrations, is extended to the contact category, and the following famous results: Arnold-Liouville theorem, Atiyah-Guillemin-Sternberg convexity of the moment map theorem, Delzant realization theorem, Duistermaat theory have been shown to admit a generalization to the contact category in the paper [6] to which this expository paper may serve as an introduction.

Then there is no non-zero G-linear map 9 ~ R~l (T*O). That is, R~l (T*O) contains no copy of the adjoint representation. 2. The formula (51) applies equally well when 9 = sp(2n, q, n 2: 1. But then P = o. The symbol fol(* x o)2 easily quantizes to a . differential operator on o. See [A-B2]. Our main result is the G-equivariant quantization of these symbols r x into differential operators Dx on 0 in the cases where 9 is classical. 3. Assume 9 is a complex simple Lie algebra of classical type and 9 =1= sp(2n, q, n 2: 1. *

The functions f;, k = 0,1,2 ... ,x. Precisely, f; generates Rk(O) under the action of n- so that U(n-) . f; = Rk(O). We have n- c go E9 gneg and also go . 1(iii). It follows that (61) Now suppose TU;) E R(O). ,Z for all Z E gneg, we get T(Rk(O)) = T(U(gneg) . f;) = U(gneg) . TU;) c R(O). 2. The converse is obvious. 4. Suppose S is a differential operator on oreg such that S is Euler homogeneous of degree and S satisfies (59). of~-l for k = 0,1,2 ... where ,0,/1, ... are scalars. Moreover (i) and (ii) imply (iii) fo1S is a lowest weight vector of a copy ofg in V_ 1(O) .

### Advances in Geometry by Alexander Astashkevich (auth.), Jean-Luc Brylinski, Ranee Brylinski, Victor Nistor, Boris Tsygan, Ping Xu (eds.)

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