Masatoshi Noumi

Representations of the quantum group SUq(2) and the little q-Jacobi polynomials

Abstract In this paper, we study the finite dimensional unitary representations of the quantum group SUq(2). Then we obtain the Peter-Weyl theorem for SUq(2) and the matrix elements of these unitary representations are explicitly expressed in terms of the little q-Jacobi polynomials which are known as q-analogues of orthogonal polynomials. Using these expressions, the orthogonality relations of these polynomials are obtained in terms of the Haar measure on the quantum group SUq(2).

Unitary representations of the quantum group SU q (1,1): Structure of the dual space ofU q (sl(2))

Real forms of the quantum universal enveloping algebraUq(sl(2)) and a topological quantum group associated with this algebra are discussed.

Unitary representations of the quantum group SUq(1, 1): II - Matrix elements of unitary representations and the basic hypergoemetric functions

Some series of unitary representations of the quantum group SUq(1, 1) are introduced. Their matrix elements are expressed in terms of the basic hypergeometric functions. Operator realization of the coordinate elements of SUq(1, 1) and aq-analogue of some classical identities are discussed.

Big q-Jacobi polynomials, q-Hahn polynomials, and a family of quantum 3-spheres

The big q-Jacobi polynomials and the q-Hahn polynomials are realized as spherical functions on a new quantum SUq(2)-space which can be regarded as the total space of a family of quantum 3-spheres.

Wronskian Determinants and the Gröbner Representation of a Linear Differential Equation: An Approach to Nonlinear Integrable Systems

Publisher Summary This chapter presents the theory of integrable systems of nonlinear differential equations. The correspondence between linear differential equations and their solutions gives rise to various types of isomorphisms between Grassmann manifolds and a theory of nonlinear integrable systems are constructed on the bridge where Grassmann manifolds of different origins are linked. The chapter discusses the basic properties of Wronskian determinants in several variables, in the category of Д-modules over a differential field K. In order to interpret the Grassmannian formalism of in explicit terms, the canonical forms of cyclic Д -modules are studied with respect to a well-ordering of the lattice Nr; this gives a generalization of the theory of Grobner bases for polynomial ideals to a noncommutative case. The chapter also provides a framework for the generalization of a restricted version of the KP hierarchy to higher-dimensional cases.

Representation of Quantum Groups

The concept of quantum groups is important for the study of the quantum Yang-Baxter equations, Drinfeld [2], Jimbo [4], Manin [5] and others. On the other hand, Woronowicz [10] introduced the concept of compact matrix pseudogroups through the study of the dual object of groups. As pointed out by Rosso in [8], these two concepts are related to each other as quantum Lie algebras and quantum Lie groups. In this talk we want to indicate that the ideas of Kac algebras studied by Takesaki [9] and Enock and Schwartz [3] et al. are helpful for the study of quantum groups. As a result we can give a geometric interpretation for a q-analogue of a certain class of special functions, which has been a long standing problem of q- analogues.

Matrix Elements of Unitary Representation of the Quantum Group SUq (1,1) and the Basic Hypergeometric Functions

In this paper, we will study matrix elements of unitary representations of the quantum group SUq (1,1). We begin with classification of real forms of the universal quantum enveloping algebra U q (sl(2)) (§1), and next consider the structure of a topological quantum group A associated with this algebra (§2). In §3, we “exponentiate” a family of infinite dimensional representations of U q(sl(2)), and determine the matrix elements in A in terms of the basic hypergeometric functions in the q-analogue analysis. The differential representation of U q (sl(2)) in A provides us a fundamental tool for this procedure. In § 4, introducing the unitary structure of the real form U q (su(1, 1)), we classify series of unitary representations of the quantum group SUq (1,1). Throughout this paper, Z denotes the set of integers, ℕ the set of non-negative integers.