Exchange Dynamical Quantum Groups

For any simple Lie algebra ? and any complex number q which is not zero or a nontrivial root of unity, %but may be equal to 1 we construct a dynamical quantum group (Hopf algebroid), whose representation theory is essentially the same as the representation theory of the quantum group U _q (?). This dynamical quantum group is obtained from the fusion and exchange relations between intertwining operators in representation theory of U _q (?), and is an algebraic structure standing behind these relations. Received: 24 March 1998 / Accepted: 14 February 1999     

Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

In this paper we determine the support of the irreducible spherical representation (i.e., the irreducible quotient of the polynomial representation) of the rational Cherednik algebra of a finite Coxeter group for any value of the parameter c. In particular, we determine for which values of c this representation is finite dimensional. This generalizes a result of Varagnolo and Vasserot (2009) [20], who classified finite dimensional spherical representations in the case of Weyl groups and equal parameters (i.e., when c is a constant function). Our proof is based on the Macdonald–Mehta integral and the elementary theory of distributions.     

Continuous Hecke algebras

The theory of PBW properties of quadratic algebras, to which this paper aims to be a modest contribution, originates from the pioneering work of Drinfeld (see [Dr1]). In particular, as we learned after publication of [EG] (to the embarrassment of two of us!), symplectic reflection algebras, as well as PBW theorems for them, were discovered by Drinfeld in the classical paper [Dr2] 15 years before [EG] (namely, they are a special case of degenerate affine Hecke algebras for a finite group G introduced in [Dr2, Section 4]).     

Quantization of Lie bialgebras, II

This paper is a continuation of [EK]. We show that the quantization procedure of [EK] is given by universal acyclic formulas and defines a functor from the category of Lie bialgebras to the category of quantized universal enveloping algebras. We also show that this functor defines an equivalence between the category of Lie bialgebras over k [[h]] and the category of quantized universal enveloping (QUE) algebras.     

Orthogonality relations and Cherednik identities for multivariable Baker–Akhiezer functions

We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type.     

When is the Fourier transform of an elementary function elementary?

Let F be a local field, a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form , where , Q is a rational function (the phase function), are polynomials, and multiplicative characters of F. For generic , this function canonically extends to a distribution on V (if char(F) = 0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject of our study. In the first part of this paper we show that for or , if the Fourier transform of an elementary function with phase function -Q such that is another elementary function with phase function , then is the Legendre transform of Q (the "semiclassical condition"). We study properties and examples of phase functions satisfying this condition, and give a classification of phase functions such that both Q and are of the form f(x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak's classification theorem for Severi varieties.? In the second part of the paper we give a necessary and sufficient condition for an elementary function to have an elementary Fourier transform (in an appropriate "weak" sense) and explicit formulas for such Fourier transforms in the case when Q and are monomials, over any local field F. We also describe a generalization of these results to the case of monomials of norms of finite extensions of F. Finally, we generalize some of the above results (including Fourier integration formulas) to the case when and Q comes from a prehomogeneous vector space.     

Representations of affine Lie algebras,ellipticr-matrix systems,and special functions

The author considers an elliptic analogue of the Knizhnik-Zamolodchikov equations—the consistent system of linear differential equations arising from the elliptic solution of the classical Yang-Baxter equation for the Lie algebra . The solutions of this system are interpreted as traces of products of intertwining operators between certain representations of the affine Lie algebra. A new differential equation for such traces characterizing their behavior under the variation of the modulus of the underlying elliptic curve is deduced. This equation is consistent with the original system.It is shown that the system extended by the new equation is modular invariant, and the corresponding monodromy representations of the modular group are defined. Some elementary examples in which the system can be solved explicitly (in terms of elliptic and modular functions) are considered. The monodromy of the system is explicitly computed with the help of the trace interpretation of solutions. Projective representations of the braid group of the torus and representations of the double affine Hecke algebra are obtained.     

Explicit quantization of dynamical r-matrices for finite dimensional semisimple Lie algebras

We provide an explicit quantization of dynamical r-matrices for semisimple Lie algebras, classified earlier by the third author, which includes the Belavin-Drinfeld r-matrices. We do so by constructing an appropriate (dynamical) twist in the tensor square of the Drinfeld-Jimbo quantum group , which twists the R-matrix of into the desired quantization. The construction of this twist is based on the method stemming from the work of Jimbo-Konno-Odake-Shiraishi and Arnaudon-Buffenoir-Ragoucy-Roche, i.e. on defining the twist as a unique solution of a suitable difference equation. This yields a simple closed formula for the twist.

This construction allows one to confirm the alternate version of the Gerstenhaber-Giaquinto-Schack conjecture (about quantization of Belavin-Drinfeld r-matrices for in the vector representation), which was stated earlier by the second author on the basis of computer evidence. It also allows one to define new quantum groups associated to semisimple Lie algebras. We expect them to have a rich structure and interesting representation theory.

     

On Flows in Porous Media with Moving Boundaries Arising as Scaling Limits of Discrete Aggregation Models

We study porous medium (or, equivalently, Hele-Shaw) flows with a moving boundary which arise as scaling (i.e., continuous) limits of certain discrete aggregation models. Specifically, we study the scaling limit of the internal DLA model with a killing and a one-way passing condition for particles on the negative semi-axis. These models were recently studied by L. Levine and Y. Peres. We give an exact self-similar solution for the corresponding porous medium flow in the first case (which matches perfectly the numerical data obtained by L. Levine and Y. Peres), and derive moment properties of such solution in the second case.