The Canonical Solutions of the Q-Systems¶ and the Kirillov–Reshetikhin Conjecture (original) (raw)

The Canonical Solutions of the Q -Systems¶ and the Kirillov-Reshetikhin Conjecture

Communications in Mathematical Physics, 2002

We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite Q-systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland-Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov-Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.

The Kirillov-Reshetikhin conjecture and solutions of T -systems

Journal Fur Die Reine Und Angewandte Mathematik, 2006

We prove the Kirillov-Reshetikhin conjecture for all untwisted quantum affine algebras : we prove that the character of Kirillov-Reshetikhin modules solve the Q-system and we give an explicit formula for the character of their tensor products. In the proof we show that the Kirillov-Reshetikhin modules are special in the sense of monomials and that their q-characters solve the T-system (functional relations appearing in the study of solvable lattice models). Moreover we prove that the T-system can be written in the form of an exact sequence. For simply-laced cases, these results were proved by Nakajima with geometric arguments which are not available in general. The proof we use is different and purely algebraic, and so can be extended uniformly to non simply-laced cases.

Irreducible representations of deformed oscillator algebra and q-special functions

International Journal of Modern Physics A, 1996

Different generators of a deformed oscillator algebra give rise to one-parameter families of qqq-exponential functions and qqq-Hermite polynomials related by generating functions. Connections of the Stieltjes and Hamburger classical moment problems with the corresponding resolution of unity for the qqq-coherent states and with 'coordinate' operators - Jacobi matrices, are also pointed out.

On the fermionc formula and the Kirillov-Reshetikhin conjecture

2000

The fermionic formula conjectured by Kirillov and Reshetikhin describes the decomposition (as a module for Uq(frakg)U_q(\frak g)Uq(frakg)) of a tensor product of multiples of of fundamental representations W(mlambdai)W(m\lambda_i)W(mlambdai) of the corresponding quantum affine algebras. In this paper, we show that the conjecture is true for the modules W(m\lambda_i), if iii is such that the corresponding simple root occurs in the

A q-Analog of the Hua Equations

2009

A necessary condition is established for a function to be in the image of a quantum Poisson integral operator associated to the Shilov boundary of the quantum matrix ball. A quantum analogue of the Hua equations is introduced.

Quantum affine algebras, canonical bases, and q -deformation of arithmetical functions

Pacific Journal of Mathematics, 2012

In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases. Suggested by these formulas, we define natural q-deformation of arithmetical functions such as (multi-)partition function and Ramanujan τ-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider q-deformation of Kostant's function and study certain q-polynomials whose special values are weight multiplicities.

Models of q-algebra representations: Matrix elements of the q-oscillator algebra

Journal of Mathematical Physics, 1993

This paper continues a study of one and two variable function space models of irreducible representations of q-analogs of Lie enveloping algebras, motivated by recurrence relations satis ed by q-hypergeometric functions. Here we consider the quantum algebra Uq (su 2 ). We show that various q-analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and we compute the corresponding matrix elements of the \group operators" on these representation spaces. This \local" approach applies to more general families of special functions, e.g., with complex arguments and parameters, than does the quantum group approach. We show that the matrix elements themselves transform irreducibly under the action of the quantum algebra. We nd an alternate and simpler derivation of a q-analog, due to Groza, Kachurik and Klimyk, of the Burchnall-Chaundy formula for the product of two hypergeometric functions 2 F 1 . It is interpreted here as the expansion of the matrix elements of a \group operator" (via the exponential mapping) in a tensor product basis in terms of the matrix elements in a reduced basis.

Infinite Dimensional Algebras and Quantum Integrable Systems

Progress in Mathematics, 2005

Bibliographic information published b; Hie Deutsche Ribliothck Die Deutsche Bihliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. ISBN 3-7643-7215-X Birkhauser Verlag, Basel-Boston-Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained.

Models of q-algebra representations: Tensor products of special unitary and oscillator algebras

Journal of Mathematical Physics, 1992

This paper begins a study of one-and two-variable function space models of irreducible representations of q analogs of Lie enveloping algebras, motivated by recurrence relations satisfied by q-hypergeometric functions. The algebras considered are the quantum algebra U,(su2) and a q analog of the oscillator algebra (not a quantum algebra). In each case a simple one-variable model of the positive discrete series of finite-and infinitedimensional irreducible representations is used to compute the Clebsch-Gordan coefficients. It is shown that various q analogs of the exponential function can be used to mimic the exponential mapping from a Lie algebra to its Lie group and the corresponding matrix elements of the "group operators" on these representation spaces are computed. It is shown that the matrix elements are polynomials satisfying orthogonality relations analogous to those holding for true irreducible group representations. It is also demonstrated that general q-hypergeometric functions can occur as basis functions in two-variable models, in contrast with the very restricted parameter values for the q-hypergeometric functions arising as matrix elements in the theory of quantum groups.