**Seminar for Geometry, education and visualization with applications **

**PROGRAM**

MATEMATICKI INSTITUT SANU

Seminar geometriju, obrazovanje i vizualizaciju sa primenama

__PLAN RADA ZA SEPTEMBAR 2004.__

** CETVRTAK, 02. septembar 2004. u 17 sati Nema sastanka seminara. **

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** CETVRTAK, 09. septembar 2004. u 17 sati Prof. dr Luc Vrancken, LAMATH, Universite de Valenciennes An elementary problem in linear algebra **

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** CETVRTAK, 16. septembar 2004. u 17 sati Nema sastanka seminara. **

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** UTORAK, 21. septembar 2004. u 17 sati Prof. dr Fabio Gavarini, Universita degli Studi di Roma "Tor Vergata" POISSON QUOTIENTS, COISOTROPIC SUBGROUPS, AND QUANTIZATIONS: A GENERALIZED QUANTUM DUALITY PRINCIPLE **

** ABSTRACT: A special class of Poisson homogeneous spaces is that of Poisson quotients, i.e. those Poisson homogeneous spaces whose symplectic foliation has at least one zero-dimensional leaf. Poisson quotient behave nicely with respect to the natural relationship between Poisson homogeneous G-spaces (G being a Poisson group) and subgroups of G: indeed, they bijectively corresponds with subgroups which are also *coisotropic submanifolds*, i.e. their defining ideal is a Poisson subalgebra. In infinitesimal terms, the coisotropy condition corresponds to its cotangent space at the unit point to be a Lie coideal (of the tangent Lie bialgebra of the group). The natural Poisson duality among Poisson groups - which "integrates" the (linear) duality endofunctor of Lie bialgebras - extends in a natural way to a duality relation between all coisotropic subgroups of a Poisson group G and all coisotropic subgroups of G^* (the dual Poisson group to G). A similar duality holds between Poisson quotients of G and Poisson quotients of G^*. When considering quantizations of a Poisson homogeneous space, one finds that these may exist only if the space is a Poisson quotient: thus the notion of Poisson quotient proves natural also from the point of view of quantization. The quantization problem for a Poisson G-quotient then corresponds to a like question for the attached coisotropic subgroup of G (or for its infinitesimal data). For Poisson groups (or Lie bialgebras), a key tool in quantization theory is the quantum duality principle. This claims that any quantization of a given Lie bialgebra provides also a quantization of the dual Poisson group, and viceversa. The purpose of this talk is to show that such a principle can be extended to coisotropic subgroups - or, correspondingly, to Poisson quotients - involving the "generalized Poisson duality" mentioned above. As an application, this principle allows to build up new quantizations out of known ones: if a quantization of some Poisson quotient (for G) is given, then a quantization of its dual Poisson quotient (for G^*) can be explicitly constructed. Time permitting, I will show how to construct a quantization of the space of Stokes matrices (upper-diagonal unipotent matrices) - as a Poisson quotient of SL(n)^*, considered by Dubrovin, Ugaglia and Boalch in the framework of moduli spaces of semisimple Frobenius manifolds - out of a well-known quantization of SO(n). **

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** Sednice seminara odrzavaju se u zgradi Srpske akademije nauka i umetnosti, Beograd, Knez Mihailova 35, na prvom spratu u sali 2.**