Localization for quantum groups at a root of unity

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dc.creatorBackelin, Erik
dc.creatorKremnizer, Kobi
dc.date2004-07-04
dc.date2006-10-03
dc.date.accessioned2026-07-07T08:42:27Z
dc.date.available2026-07-07T08:42:27Z
dc.descriptionIn the paper \cite{BK} we 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 category 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 an open subset ofthe 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 \cite{BMR} similar results were obtained for a Lie algebra $\g_p$ in char $p$. Hence, representations of $\g_p$ 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.
dc.descriptionMistakes corrected. Added content
dc.identifierhttps://arxiv.org/abs/math/0407048
dc.identifierhttp://arxiv.org/abs/math/0407048
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/141964
dc.subjectRepresentation Theory
dc.subjectQuantum Algebra
dc.titleLocalization for quantum groups at a root of unity
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