arXiv subjects
Publications and source records attributed to Kobi Kremnizer.
We prove a singular version of Beilinson-Bernstein localization for a complex semi-simple Lie algebra following ideas from the positive characteristic case done by \cite{BMR2}. We apply this theory to translation functors, singular blocks in the Bernstein-Gelfand-Gelfand category $\BGGcat$ and Whittaker modules.
We quantize parabolic flag manifolds and describe categories of equivariant quantum $\D$-modules on them at a singular central character. We compute global sections at any $q \in \C^*$ and we also prove a singular version of Beilinson-Bernstein localization for a quantized enveloping algebra $\Uq(\g)$, when $q$ is generic.
We give a short new proof for the theorem that global sections of the sheaf of quantum differential operators on a quantum flag manifold are given by the quantum group. As corollaries we retrieve Joseph and Letzter's quantum versions of classical enveloping algebra theorems of Duflo and Kostant. We also describe the center of the ad-integrable part of the quantum group and the adjoint Lie algebra action on it.
We construct symmetric monoidal categories $\LRF, \FD$ of rooted forests and Feynman graphs. These categories closely resemble finitary abelian categories, and in particular, the notion of Ringel-Hall algebra applies. The Ringel-Hall Hopf algebras of $\LRF, \FD$, $\HH_{\LRF}, \HH_{\FD}$ are dual to the corresponding Connes-Kreimer Hopf algebras on rooted trees and Feynman graphs. We thus obtain an interpretation of the Connes-Kreimer Lie algebras on rooted trees and Feynman graphs as Ringel-Hall Lie algebras.
We construct a central extension of the group of automorphisms of a 2-Tate vector space viewed as a discrete 2-group. This is done using an action of this 2-group on a 2-gerbe of gerbel theories. This central extension is used to define central extensions of double loop groups.
In this paper we prove a conjecture by De Concini, Kac and Procesi \cite{CP} (Corollary \ref{conj}): The dimension of any $M\in U_q-\mood^χ$ is divisible by $l^{codim_\mathcal{B}\mathcal{B}_χ}$.
In 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.
Let $\Oq(G)$ be the algebra of quantized functions on an algebraic group $G$ and $\Oq(B)$ its quotient algebra corresponding to a Borel subgroup $B$ of $G$. We define the category of sheaves on the "quantum flag variety of $G$" to be the $\Oq(B)$-equivariant $\Oq(G)$-modules and proves that this is a proj-category. We construct a category of equivariant quantum $\mathcal{D}$-modules on this quantized flag variety and prove the Beilinson-Bernsteins localization theorem for this category in the case when $q$ is not a root of unity.