Youichi Shibukawa

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.

Dynamical Yang-Baxter maps

In this work, we propose and investigate dynamical YangBaxter maps, some of which produce solutions to the quantum dynamical Yang-Baxter equation. Suppose that L is a loop and a group. If their unit elements coincide, then L gives birth to a bijective dynamical Yang-Baxter map from L×L to L×L whose dynamical parameter belongs to L. The above group L is abelian if and only if the corresponding dynamical Yang-Baxter map satisfies the unitary condition.

Vertex-IRF correspondence and factorized

As for an ellipticR-operator which satisfies the Yang-Baxter equation, the incoming and outgoing intertwining vectors are constructed, and the vertex-IRF correspondence for the ellipticR-operator is obtained. The Boltzmann weights of the corresponding IRF model satisfy the star-traingle relation. By means of these intertwining vectors, the factorizedL-operators for the ellipticR-operator are also constructed. The vertex-IRF correspondence and the factorizedL-operators for BelavinsR-matrix are reproduced from those of the ellipticR-operator.

L

In this work, we present a vertex–face correspondence between an elliptic R-operator and Boltzmann weights related to the Lie superalgebra sl(mn).

-operators for an elliptic

We classified the R-operators which satisfy the quantum Yang–Baxter equation on a function space. In this study, we gave all the meromorphic solutions of the system of the functional equations which is a necessary and sufficient condition for the R-operator to satisfy the Yang–Baxter equation. Most of the solutions were expressed in terms of the elliptic, trigonometric and rational functions.

R

A q-analogue of the Frobenius formula is proved by means of the quantum groups Uq(gln+1), Aq(GLn+1) and Iwahoris Hecke algebra of type AN-1, and then, the character table of this Hecke algebra is investigated.