In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval (0, s) at the hard edge contains k eigenvalues, was evaluated in terms of a Painlevé V transcendent in σ-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this...

We extend Yangian double to super (or graded) case and give its Drinfeld generators realization by Gauss decomposition. Quantum algebras(in general meaning, it includes Yangians and quantum affine algebras etc.) are new algebraic structures discovered about ten years ago [1-4], they play important roles in the study of soluble statistical models and quantum fields theory. Quantum universal enve...

We generalize Harish-Chandra-Itzykson-Zuber and certain other integrals (Gross-Witten integral and integrals over complex matrices) using the notion of tau function of matrix argument. In this case one can reduce matrix integral to the integral over eigenvalues, which in turn is certain tau function. Soliton theory. KP hierarchy of integrable equations, which is the most popular example, consis...

Let G be a simple complex classical group and g its Lie algebra. Let U~(g) be the Drinfeld-Jimbo quantization of the universal enveloping algebra U(g). We construct an explicit U~(g)-equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers.

We suggest a possible programme to associate geometric “flaglike” data to an arbitrary simple quantum group, in the spirit of the noncommutative algebraic geometry developed by Artin, Tate, and Van den Bergh. We then carry out this programme for the standard quantum SL(n) of Drinfeld and Jimbo, where the varieties involved are certain T -stable subvarieties of the (ordinary) flag variety.

Recently Fayers introduced a large family of combinatorial realizations of the fundamental crystal B(Λ0) for b sln, where the vertices are indexed by certain partitions. He showed that special cases of this construction agree with the Misra-Miwa realization and with Berg’s ladder crystal. Here we show that another special case is naturally isomorphic to a realization using Nakajima’s monomial c...

We introduce the concept of homotopy equivalence for Hopf Galois extensions and make a systematic study of it. As an application we determine all H-Galois extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo quantum group.