2004/07/04 by Erik Backelin, Backelin, Erik, Kobi Kremnizer +1
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT
paper · pdf · doi:10.48550/arxiv.math/0407048
Mistakes corrected. Added content
openalex publication_date 2004/07/04 · arxiv created 2006/10/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the paper \citeBK we defined categories of equivariant quantum Oq-modules and Dq-modules on the quantum flag variety of G. We proved that the Beilinson-Bernstein localization theorem holds at a generic q. Here we prove that a derived version of this theorem holds at the root of unity case. Namely, the global section functor gives a derived equivalence between category of Uq-modules and Dq-modules on the quantum flag variety. For this we first prove that Dq is an Azumaya algebra over an open subset ofthe cotangent bundle T^⋆ X of the classical (char 0) flag variety X. This way we get a derived equivalence between representations of Uq and certain OT^⋆ X-modules. In the paper \citeBMR similar results were obtained for a Lie algebra \gp in char p. Hence, representations of \gp and of Uq (when q is a p'th root of unity) are related via the cotangent bundles T^⋆ X in char 0 and in char p, respectively.