Kimio Ueno

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.

Completely ℤ symmetric R matrix

An infinite-dimensional R matrix related to the limiting case n→∞ of the completely ℤn symmetric R matrix is discovered. This R matrix is expressed as an operator on C∞(S1×S1). Moreover, the fusion procedure of the R-operator is investigated and the finite-dimensional R matrices are constructed from the R operator.

Gelfand-Zetlin basis for Uq(gl(N+1)) modules

The Gelfand-Zetlin basis of Uq(gl(N+1)) modules is constructed via the lowering operator method.

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.

The Inversion Formula of Polylogarithms and the Riemann-Hilbert Problem

In this article, we set up a method of reconstructing the polylogarithms \(\operatorname {Li}_{k}(z)\) from zeta values ζ(k) via the Riemann-Hilbert problem. This is referred to as “a recursive Riemann-Hilbert problem of additive type.” Moreover, we suggest a framework of interpreting the connection problem of the Knizhnik-Zamolodchikov equation of one variable as a Riemann-Hilbert problem.

KZ equation on the moduli space ℳ0,5 and the harmonic product of multiple polylogarithms

In this article, we derive a system of functional relations called the generalized harmonic product relations for hyperlogarithms on the moduli space

CHARACTER TABLE OF HECKE ALGEBRA OF TYPE AN-1 AND REPRESENTATIONS OF THE QUANTUM GROUP Uq(gln+1)

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Representation of Quantum Groups

and show that the relations contain the harmonic product of multiple polylogarithms. The generalized harmonic product relations are equivalent to the relations which come from two decompositions of the fundamental solution normalized at the origin of the KZ equation on

Spectrum of an operator appears in the quantum SU(1,1) group

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