Geometry of the Pfaff Lattices (original) (raw)

The (semi-infinite) Pfaff lattice was introduced by Adler and van Moerbeke [2] to describe the partition functions for the random matrix models of GOE and GSE type. The partition functions of those matrix models are given by the Pfaffians of certain skew-symmetric matrices called the moment matrices, and they are the τ -functions of the Pfaff lattice. In this paper, we study a finite version of the Pfaff lattice equation as a Hamiltonian system. In particular, we prove the complete integrability in the sense of Arnold-Liouville, and using a moment map, we describe the real isospectral varieties of the Pfaff lattice. The image of the moment map is a convex polytope whose vertices are identified as the fixed points of the flow generated by the Pfaff lattice.