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 | References | Similar Articles | Additional Information

Abstract: In the paper *Quantum flag varieties, equivariant quantum -modules, and localization of Quantum groups*, Backelin and Kremnizer defined categories of equivariant quantum -modules and -modules on the quantum flag variety of . We proved that the Beilinson-Bernstein localization theorem holds at a generic . 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 -modules and -modules on the quantum flag variety.

For this we first prove that is an Azumaya algebra over a dense subset of the cotangent bundle of the classical (char 0) flag variety . This way we get a derived equivalence between representations of and certain -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 in char . Hence, representations of and of (when is a 'th root of unity) are related via the cotangent bundles in char 0 and in char , respectively.

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Additional Information

**Erik Backelin**

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

Email:
erbackel@uniandes.edu.co

**Kobi Kremnizer**

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

Email:
kremnize@math.mit.edu

DOI:
https://doi.org/10.1090/S0894-0347-08-00608-5

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.