The Pfaff lattice was introduced by us in the context of a Lie
algebra splitting of ${\rm gl}(\infty)$ into ${\rm sp}(\infty)$ and an
algebra of lower-triangular matrices. The Pfaff lattice is equivalent
to a set of bilinear identities for the wave functions, which yield
the existence of a sequence of "$\tau$-functions". The latter satisfy
their own set of bilinear identities, which moreover characterize
them.
¶ In the semi-infinite case, the $\tau$-functions are Pfaffians, in
the same way that for the Toda lattice the $\tau$-functions are
Hänkel determinants; interesting examples occur in the theory
of random matrices, where one considers symmetric and symplectic
matrix integrals for the Pfaff lattice and Hermitian matrix integrals
for the Toda lattice.
¶ There is a striking parallel between the Pfaff lattice and the Toda
lattice, and even more striking, there is a map from one to the other,
mapping skew-orthogonal to orthogonal polynomials. In particular, we
exhibit two maps, dual to each other, mapping Hermitian matrix
integrals to either symmetric matrix integrals or symplectic matrix
integrals.