Poisson geometry of the analog of the Miura maps and Bäcklund-Darboux transformations for equations of Toda type and periodic Toda flows (Q1182208)

It is shown that the analog of the Miura maps and the Bäcklund-Darboux transformations for systems of Toda type on real, semisimple Lie groups are isomorphisms of Poisson Lie groups. As an application of these results in the study of the interpolation of discrete algorithms in numerical linear algebra by continuous flows, the complete integrability of the so-called \(SVD\) flow on the upper triangular group is established. The same is done for the generalized periodic Toda flows, where the dynamics now takes place on appropriate infinite-dimensional loop groups. Finally, the classical periodic Toda lattice is considered. Here, the maps which take the \(2n\)-periodic Kac-van Moerbeke lattice to the \(n\)- periodic Toda lattice are shown to be Poisson maps.