Generalized Toda Mechanics Associated with Loop Algebras (L)(Cr) and (L)(Dr) and Their Reductions

We construct a class of integrable generalization of Todamechanics with long-range interactions. These systems areassociated with the loop algebras L(C_r) and L(D_r) in the sense that their Lax matrices can be realized interms of the c=0 representations of the affine Lie algebrasC⁽¹⁾_r and D⁽¹⁾_r and the interactions pattern involvedbears the typical characters of the corresponding root systems. Wepresent the equations of motion and the Hamiltonian structure.These generalized systems can be identified unambiguously byspecifying the underlying loop algebra together with an orderedpair of integers (n,m). It turns out that different systemsassociated with the same underlying loop algebra but withdifferent pairs of integers (n₁,m₁) and (n₂,m₂) withn₂<n₁ and m₂<m₁ can be related by a nested Hamiltonianreduction procedure. For all nontrivial generalizations, the extracoordinates besides the standard Toda variables are Poissonnon-commute, and when either $n$ or m≥3, the Poissonstructure for the extra coordinate variables becomes some Liealgebra (i.e. the extra variables appear linearly on theright-hand side of the Poisson brackets). In the quantum case, suchgeneralizations will become systems with noncommutative variableswithout spoiling the integrability.