Discrete Integrable Systems and the Moyal Symmetry

In the study of nonlinear systems, completely integrable systems play special roles. They are not independent at all, but are strongly correlated with each other owing to the large symmetries shared among themselves. We like to know how far we can extend such systems without loosing integrability. The question could be answered if we know how much we can deform the symmetries characterizing the integrable systems. In this note, we would like to discuss the Moyal bracket algebra [1], a quantum deformation of the Poisson bracket algebra, as a scheme which should describe a large class of integrable systems. Here, however, we focus our attention to the Hirota bilinear difference equation (HBDE) [2], a difference analogue of the two-dimensional Toda lattice. This paper is organized as follows. We first explain HBDE and what is the Moyal quantum algebra in the following two sections. Using the Miwa transformation [3] of soliton variables, the correspondence of the shift operator appeared in HBDE to the Moyal quantum operator is shown in sections 4 and 5. The large symmetry possessed by the universal Grassmannian of the KP hierarchy [4, 5] is explained in section 6 within our framework, and the connection of their generators to the Moyal quantum operators is shown in the last section.