Lie algebraic aspects of the finite nonperiodic Toda flows

In this paper we review various aspects of nonperiodic full and tridiagonal Toda flow theory. In particular, we consider gradient structures in the dynamics and geometry of these systems and we compare and contrast a number of formulations of the nonperiodic Toda equations. We describe the structure of these systems on general semisimple Lie algebras. In the case of the full Kostant (asymmetric) Toda flow we describe some of the background behind its integrability and explain the role of noncommutative integrability in its qualitative behavior. We describe the relationship between the asymmetric Toda flows and the symmetric and indefinite Toda flows, and show how one may conjugate from the full Kostant Toda flows to the full symmetric Toda flows via a Poisson map. We describe briefly related Toda systems such as nonabelian Toda, and the peakon flows.