Mathematics of Symmetry

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The main theme of this project is the symmetry approach to solvable lattice models. As for this problem, In the case of elliptic models, which had not been solved In the symmetric approach, a bosonization of the vertex operators were obtained by Shiraishi and Odake using the representation theory of quasl-Hopf algebra, in particular, the twist of quantum groups. By Miwa and Konno, in the trigonometric limit (with |q |=1) of this model, an integral formula is obtained for the difference analogue of the Knnizhnik-Zamolodchhikov equation. Hasegawa constructed Ruijsenaars' commuting difference operators by using the intertwining vectors In the elliptic model.The theory of crystal is a key In the connection between solvable lattice models and combinatorics. As for this, Miwa, Okado, Kuniba, Yamada (Yasuhiko) found that the set of inhomogeneous paths give the crystal of tensor products of integrable highest weight representations. Several interesting examples of non-perfect crystals and the … More corresponding paths are also studied.As for non-solvable models, Matsui showed that the matrix product states which represent the ground states correspond to the representations of the Kuntz algebra.The toroidal algebra is important because it governs the symmetry of the vertex operators. Mild found an automorphism of this algebra which connects two affine quantum algebras inside thereof.Solvable models In quantum field theory Is another main subject. Kawahigashi developed the method for calculating modular invariant quantities in the two-dimensional conformal field theory. It is also important to apply techniques developed in the two-dimensional solvable models to the problems in string theory and the four-dimensional gauge theory. As for this, Nakatsu showed that in the adiabatic limit the tau function of the Toda lattice gives the effective action of the N=2 super-symmetric Yang-Mills theory. Kanno and Yang obtained a generalization of the Donaldson-Witten Invariant for the four dimensional manifolds. Kato analysed the condition for a matrix model to be understood as quantum gravity theory In a curved space-time.The developments In the past three years Include(1) the symmetry method for the solvable lattice models of elliptic type ;(2) the method of representation theory for the mixed spin chains ;(3) the discovery of new links between lattice models and combinatorics ;(4) the quantization of several geometric invariants ;(5) the algebraic approach connecting the operator algebras and the conformal field theory. Less