Deforming commutative algebras in the lower triangular (ℤ×ℤ)-matrices yields lower triangular Toda hierarchies and their associated nonlinear equations. Like for their counterpart in the ring of pseudodifferential operators, the KP-hierarchy, one also has for these hierarchies a geometric picture: certain infinite chains of subspaces in an separable Hilbert space provide solutions of lower triangular Toda hierarchies. The KP-hierarchy and its multi-component version contain many interesting subsystems, like e.g. the nth Gelfand–Dickey hierarchy and the AKNS-hierarchy. In this paper one considers analogues of these two subsystems in the context of the lower triangular Toda hierarchies and a geometric description of solutions to both type reductions is given.
- nth Gelfand–Dickey hierarchy - AKNS-hierarchy - Finite band deformation - Commuting flows - Banach Lie group - Big cell
- Lower triangular Toda hierarchy - Lax equations - Linearization - Reduction - KP-hierarchy -