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Localization for quantum groups at a root of unity

Authors: Erik Backelin and Kobi Kremnizer
Journal: J. Amer. Math. Soc. 21 (2008), 1001-1018
MSC (2000): Primary 14A22, 17B37, 58B32; Secondary 20G42
Published electronically: June 19, 2008
MathSciNet review: 2425178
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Abstract: In the paper Quantum flag varieties, equivariant quantum $ \mathcal{D}$-modules, and localization of Quantum groups, Backelin and Kremnizer defined categories of equivariant quantum $ \mathcal{O}_q$-modules and $ \mathcal{D}_q$-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 categories of $ U_q$-modules and $ \mathcal{D}_q$-modules on the quantum flag variety.

For this we first prove that $ \mathcal{D}_q$ is an Azumaya algebra over a dense subset of the cotangent bundle $ T^\star X$ of the classical (char 0) flag variety $ X$. This way we get a derived equivalence between representations of $ U_q$ and certain $ \mathcal{O}_{T^\star X}$-modules.

In the paper Localization for a semi-simple Lie algebra in prime characteristic, by Bezrukavnikov, Mirkovic, and Rumynin, similar results were obtained for a Lie algebra $ \mathfrak{g}_p$ in char $ p$. Hence, representations of $ \mathfrak{g}_p$ and of $ U_q$ (when $ q$ is a $ p$'th root of unity) are related via the cotangent bundles $ T^\star X$ in char 0 and in char $ p$, respectively.

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Additional Information

Erik Backelin
Affiliation: Departamento de Matemáticas, Universidad de Los Andes, Carrera 4, 26-51, Bogota, Colombia

Kobi Kremnizer
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

Keywords: Quantum groups, roots of unity, localization, derived equivalence, Calabi-Yau categories, noncommutative algebraic geometry
Received by editor(s): November 1, 2006
Published electronically: June 19, 2008
Additional Notes: The second author was supported in part by NSF grant DMS-0602007
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.