STUDY OF DISCRETE TIME EVOLUTIONS IN DISCRETE QUANTUM INTEGRABLE SYSTEMS

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The aim of this project is to study the integrable systems in mathematical physics, especially the relationship between those systems in discretized quantum system and the elliptic solutions of the Yang-Baxter equations.In particular, we have been concentrated on the study related to the main isuue in this subject which is to clarify the quantum group like structures and the discrete time evolution structures appeared in Sklyanin and others' work on Calogero type systems.The result is as follows. Hasegawa obtained a quantization of the Weyl group action in discrete Painleve system and the generalization thereof studied by Noumi and Yamada. The solution of the Yang-Baxter equation known as the Chiral-Potts model plays the role of quantum generating function of Noumi - Yamada - Kajiwara's canonical transformation or the discrete time evolution. We remark that the invotutivity of the transformations fails in this quantum case and generate a braid group action.These findings connect the Chiral Potts models and the Painleve equations in an unexpected way so that the mutual relationship will help to clarify the nature of these objects.On the other hand, it is well known that Panlev'e type equations can be obtained by reductions of the KP hierachy. Kuroki tried to extend Hasegawa's result of quantizing the Paileve systems via the Poisson structures. Also he studied the conformal block of a special kind over the elliptic curves, as well as studied a higher dimensional generalization of the integrable systems.