The Pfaff lattice, Matrix integrals and a map from Toda to Pfaff

摘要

We study the Pfaff lattice, introduced by us in the context of a Lie algebrasplitting of gl(infinity) into sp(infinity) and lower-triangular matrices. Weestablish a set of bilinear identities, which we show to be equivalent to thePfaff Lattice. In the semi-infinite case, the tau-functions are Pfaffians;interesting examples are the matrix integrals over symmetric matrices(symmetric matrix integrals) and matrix integrals over self-dual quaternionicHermitean matrices (symplectic matrix integrals). There is a striking parallel of the Pfaff lattice with the Toda lattice, andmore so, there is a map from one to the other. In particular, we exhibit twomaps, dual to each other, (i) from the the Hermitean matrix integrals to the symmetric matrixintegrals, and (ii) from the Hermitean matrix integrals to the symplectic matrix integrals. The map is given by the skew-Borel decomposition of a skew-symmetricoperator. We give explicit examples for the classical weights.