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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)

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)

This article is the third part of the series of papers on quantization of Lie bialgebras which we started in 1995. However, its object of study is much less general than in the previous two parts. While in the first and second paper we deal with an arbitrary Lie bialgebra, here we study Lie bialgebras of g-valued functions on a punctured rational or… (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)

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)

We study the Galois group of a matrix q-difference equation with rational coefficients which is regular at 0 and ∞, in the sense of (difference) Picard-Vessiot theory, and show that it coincides with the algebraic group generated by matrices C(u)C(w)−1 u,w ∈ C∗ , where C(z) is the Birkhoff connection matrix of the equation. 1. Differential algebra The… (More)

We construct an endoscopic decomposition for local L-packets associated to irreducible cuspidal Deligne–Lusztig representations. Moreover, the obtained decomposition is compatible with inner twistings.

- David Kazhdan, A . POLISHCHUK
- 2004

In the first part of this paper we study minimal representations of simply connected simple split groups G of type Dk or Ek over local nonarchimedian 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 to… (More)