We study tensor products of the spin modules (i.e. the Fermion Fock space representations) for classical (simple or affine) Kac-Moody Lie algebras. We find out that there are mutually commutant pairs of classical Kac-Moody algebras acting on the spin modules, and describe the irreducible decompositions in terms of Young diagrams. As applications, we obtain a simple explanation of Jimbo-Miwa's branching rule duality (i.e. isomorphisms between coset Virasoro modules) [JM], generalization thereof and the duality of the modular transformation rules of affine Lie algebra characters.
The super-Toda lattice (STL) hierarchy is introduced. The equivalence between the Lax representation and Zakharov-Shabat representation of the STL hierarchy is shown. Introducing the Lie superalgebra osp(∞|∞), the ortho-symplectic (OSp)-STL hierarchy is defined as well. These equations are solved through the Riemann-Hilbert decomposition of corresponding infinite dimensional Lie supergroups. An explicit representation of solutions is given by means of the super-τ field.