On the Liouville Integrability of the Periodic Kostant–Toda Flow on Matrix Loops of Level k

In this work, we consider the periodic Kostant–Toda flow on matrix loops in \({\mathfrak{s}\mathfrak{l}(n,\mathbb{C})}\) of level k, which correspond to periodic infinite band matrices with period n with lower bandwidth equal to k and fixed upper bandwidth equal to 1 with 1’s on the first superdiagonal. We show that the coadjoint orbits through the submanifold of such matrix loops can be identified with those of a finite-dimensional Lie group, which appears in the form of a semi-direct product. We then characterize the generic coadjoint orbits and obtain an explicit global cross-section for such orbits. We also establish the Liouville integrability of the periodic Kostant–Toda flow on such orbits via the construction of action-angle variables.