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The working mathematician fears complicated words but loves pictures and diagrams. We thus give a no-fancy-anything picture rich glimpse into Khovanov's novel construction of " the categorification of the Jones polynomial ". For the same low cost we also provide some computations, including one that shows that Khovanov's invariant is strictly stronger than… (More)

This paper is a continuation of [EK1-4]. The goal of this paper is to define and study the notion of a quantum vertex operator algebra (VOA) in the setting of the formal deformation theory and give interesting examples of such algebras. Our definition of a quantum VOA is based on the ideas of the paper [FrR]. The first chapter of our paper is devoted to the… (More)

We develop a theory of integration over valued fields of residue characteristic zero. In particular we obtain new and base-field independent foundations for integration over local fields of large residue characteristic, extending results of Denef,Loeser, Cluckers. The method depends on an analysis of definable sets up to definable bijections. We obtain a… (More)

In the paper [Dr3] V.Drinfeld formulated a number of problems in quantum group theory. In particular, he raised the question about the existence of a universal quantization for Lie bialgebras, which arose from the problem of quantization of Poisson Lie groups. When the paper [KL] appeared Drinfeld asked whether the methods of [KL] could be useful for the… (More)

Theta series for exceptional groups have been suggested as a possible description of the eleven-dimensional quantized BPS membrane. We present explicit formulae for these automorphic forms whenever the underlying Lie group G is simply laced. Specifically, we review and construct explicitly the minimal representation of G which generalizes the Schrödinger… (More)

- D Kazhdan, A Polishchuk
- 2004

In the first part of this paper we study minimal representations of simply connected simple split groups G of type D k or E k over local non-archimedian fields. Our main result is an explicit formula for the spherical vectors in these representations. In the case of groups over R and C, such a formula was obtained recently in [8]. We also use our techniques… (More)

We study the Galois group of a matrix q-diierence equation with rational coeecients which is regular at 0 and 1, in the sense of (diierence) Picard-Vessiot theory, and show that it coincides with the algebraic group generated by matrices C(u)C(w) ?1 u; w 2 C , where C(z) is the Birkhoo connection matrix of the equation. The notion of the Galois group of a… (More)

- David Kazhdan, Dennis Gaitsgory
- 2008

We introduce a categorical framework for the study of representations of G F , where G is a reductive group, and F is a 2-dimensional local field, i.e. F = K((t)), where K is a local field. Our main result says that the space of functions on G F , which is an object of a suitable category of representations of G F with the respect to the action of G on… (More)

- Tomas Recio, Daniel Quillen, +11 authors Martin Raussen
- 2012

- J Bernstein, P Deligne, D Kazhdan
- 1986

w Statement of the theorem 1.1. Let G be a reductive p-adic group. A smooth representation (~', E) of the group G on a complex vector space E is called a G-module. Usually we shorten the notation and write w or E. Let d~(G) be the category of G-modules, Irr G the set of equivalence classes of irreducible G-modules, and R (G) the Grothendieck group of… (More)