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

This publication is an outgrowth of the actions of the guts for Geometry and Mathematical Physics (CGMP) at Penn kingdom from 1996 to 1998. the guts was once created within the arithmetic division at Penn kingdom within the fall of 1996 for the aim of marketing and aiding the actions of researchers and scholars in and round geometry and physics on the collage. The CGMP brings many viewers to Penn nation and has ties with different learn teams; it organizes weekly seminars in addition to annual workshops The e-book includes 17 contributed articles on present examine themes in numerous fields: symplectic geometry, quantization, quantum teams, algebraic geometry, algebraic teams and invariant thought, and personality­ istic sessions. many of the 20 authors have talked at Penn nation approximately their study. Their articles current new effects or talk about attention-grabbing perspec­ tives on contemporary paintings. the entire articles were refereed within the ordinary style of fine clinical journals. Symplectic geometry, quantization and quantum teams is one major topic of the booklet. a number of authors research deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting transverse polarizations, and reviews the instant map in relation to semisimple coadjoint orbits. Bieliavsky constructs an specific star-product on holonomy reducible sym­ metric coadjoint orbits of an easy Lie team, and he indicates easy methods to con­ struct a star-representation which has attention-grabbing holomorphic properties.

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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.

7. We will show in [A-B3] that Dx = -Da(x). 3. Strategy for quantizing the symbol roo Here is our strategy for quantizing the symbol ro into the operator Do. We start from the formula (51) for the symbol roo This says that S = foro where (58) Our idea is to construct Do by first constructing a suitable quantization S of S which is left divisible by fo, and then putting Do = folS. x ~xo of the vector fields ~ Xi , ~ x: ""' E 'I1XO on oreg . Hence S belongs to R(T*oreg). 1, we know that S belongs to R(T*O).

However, our quantization is considerably more subtle as our principal symbol TO involves a homogeneous quartic polynomial in the symbols of non-commuting vector fields (which lie outside the t-action). The quantization then requires not only symmetrization but also the introduction of lower order correction terms. 3 says that these correction terms are uniquely determined. In [1-0j, Lecomte and Ovsienko show that the algebra of polynomial symbols on lRn admits a unique quantization (involving non-obvious lower order terms) equivariant under the vector field action of sl(n+ 1, lR) on lRn 50 ALEXANDER ASTASHKEVICH AND RANEE BRYLINSKI which arises by embedding IR n as the "big cell" in IRlpm.

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