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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Localization for quantum groups at a root of unity
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by Erik Backelin and Kobi Kremnizer PDF
J. Amer. Math. Soc. 21 (2008), 1001-1018 Request permission

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
  • 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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 21 (2008), 1001-1018
  • MSC (2000): Primary 14A22, 17B37, 58B32; Secondary 20G42
  • DOI: https://doi.org/10.1090/S0894-0347-08-00608-5
  • MathSciNet review: 2425178