Kobi Kremnizer

Localization for quantum groups at a root of unity

Let C be the field of complex numbers and fix q G C*. Let g be a semi-simple Lie algebra over C and let G be the corresponding simply connected algebraic group. Let Uq be the quantized enveloping algebra of q. Let Oq be the algebra of quantized functions on G. Let Oq(B) be the quotient Hopf algebra of Oq corresponding to a Borel subgroup B of G. In the paper [BK] we defined categories of equivariant quantum O^-modules and Pg-modules on the quantum flag variety of G. We proved that the Beilinson Bernstein localization theorem holds at a generic q. Namely, the global section functor gives an equivalence between categories of [/^-modules and Vq-iaod\?es on the quantum flag variety. Thus one can translate questions about the represen tation theory of quantum groups to the study of the (noncommutative) geometry of the quantum flag variety. In this paper we prove that a derived version of this theorem holds at the root of unity case. Using this equivalence, we get that one can understand the representation theory of quantum groups at roots of unity through the (now-commutative) geometry of the Springer fibers. We now recall the main results in [BK]. The constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section. We defined an equivariant sheaf of quasi-coherent modules over the quantum flag variety to be a left Oq-mod\?e equipped with a right ?q(B)-comod\ile structure satisfying certain compatibility conditions. Such objects form a category denoted Ai?q(Gq). It contains certain line bundles Oq(\) for ? in the weight lattice. We proved that Oq(X) is ample for ? >> 0 holds for every q. This implies that the category M.Bq(Gq) is a Proj-category in the sense of Serre.

Feynman Graphs, Rooted Trees, and Ringel-Hall Algebras

We construct symmetric monoidal categories

Beyond perturbation 1: De Rham spaces

{\mathcal{LRF}, \mathcal{LFG}}

Quantum flag varieties, equivariant quantum D-modules, and localization of quantum groups

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

\mathfrak{g}

{\mathcal{LRF}, \mathcal{LFG}}

Global quantum differential operators on quantum flag manifolds, theorems of Duflo and Kostant

,