Generalized Toda Mechanics Associated with Classical Lie Algebras and Their Reductions

For any classical Lie algebra $mathfrak{g}$,we construct a family ofintegrable generalizations of Toda mechanics labeled a pair of orderedintegers $(m,n)$. The universal form of the Lax pair, equations of motion,Hamiltonian as well as Poisson brackets are provided, and explicit examplesfor $mathfrak{g}=B_{r},C_{r},D_{r}$ with $m,nleq3$ are also given. For all$m,n$, it is shown that the dynamics of the $(m,n-1)$- and the $(m-1,n)$-Todachains are natural reductions of that of the $(m,n)$-chain, and for $m=n$,there is also a family of symmetrically reduced Toda systems, the$(m,m)_{mathrm{Sym}}$-Toda systems, which are also integrable. In the quantumcase, all $(m,n)$-Toda systems with $m>1$ or $n>1$ describe the dynamics ofstandard Toda variables coupled to noncommutative variables. Except for thesymmetrically reduced cases, the integrability for all $(m,n)$-Toda systemssurvive after quantization.