Advances in Geometry: Volume 1

Description

This book is an outgrowth of the activities of the Center for Geometry and Mathematical Physics (CGMP) at Penn State from 1996 to 1998. The Center was created in the Mathematics Department at Penn State in the fall of 1996 for the purpose of promoting and supporting the activities of researchers and students in and around geometry and physics at the university. The CGMP brings many visitors to Penn State and has ties with other research groups; it organizes weekly seminars as well as annual workshops The book contains 17 contributed articles on current research topics in a variety of fields: symplectic geometry, quantization, quantum groups, algebraic geometry, algebraic groups and invariant theory, and character­ istic classes. Most of the 20 authors have talked at Penn State about their research. Their articles present new results or discuss interesting perspec­ tives on recent work. All the articles have been refereed in the regular fashion of excellent scientific journals. Symplectic geometry, quantization and quantum groups is one main theme of the book. Several authors study deformation quantization. As­ tashkevich generalizes Karabegov's deformation quantization of Kahler manifolds to symplectic manifolds admitting two transverse polarizations, and studies the moment map in the case of semisimple coadjoint orbits. Bieliavsky constructs an explicit star-product on holonomy reducible sym­ metric coadjoint orbits of a simple Lie group, and he shows how to con­ struct a star-representation which has interesting holomorphic properties Front Matter....Pages i-xi On Karabegov’s Quantizations of Semisimple Coadjoint Orbits....Pages 1-18 Exotic Differential Operators on Complex Minimal Nilpotent Orbits....Pages 19-51 The Geometry Surrounding the Arnold-Liouville Theorem....Pages 53-69 Symmetric Spaces and Star Representations....Pages 71-82 Hyperplane Arrangements, Springer Representations and Exponents....Pages 83-93 Comparison of the Beilinson-Chern Classes With the Chern-Cheeger-Simons Classes....Pages 95-105 Geometric Construction of Quillen Line Bundles....Pages 107-146 Quadratic Algebras, Dunkl Elements, and Schubert Calculus....Pages 147-182 Logarithmic Forms with Twisted Coefficients....Pages 183-193 Higher Holonomies, Geometric Loop Groups and Smooth Deligne Cohomology....Pages 195-235 Moduli Spaces of Linkages and Arrangements....Pages 237-270 Moduli Spaces of Flat Connections on 2-Manifolds, Cobordism, and Witten’s Volume Formulas....Pages 271-295 A Rigidity Property for Quantum SU (3) Groups....Pages 297-336 On the Cohomology Ring of an Algebra....Pages 337-370 On a Quantum Version of Pieri’s Formula....Pages 371-383 Some Non-Koszul Algebras....Pages 385-389 A Variety of Solutions to the Yang-Baxter Equation....Pages 391-399 Back Matter....Pages 401-403