Localizations for construction of quantum coset spaces
Abstract
Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. Particularily, in the "quantum" case, calculations with quantum minors yield the structure theorems.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 January 2003
 arXiv:
 arXiv:math/0301090
 Bibcode:
 2003math......1090S
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 14A22;
 16W30;
 14L30;
 58B32
 EPrint:
 34 pages bcp.sty