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

摘要

We construct a class of integrable generalization of Toda mechanics with long-range interactions. These systems are associated with the loop algebras L(Cr) and L(Dr) in the sense that their Lax matrices can be realized in terms of the c = 0 representations of the affine Lie algebras Cr(1) and Dr(1) and the interactions pattern involved bears the typical characters of the corresponding root systems. We present the equations of motion and the Hamiltonian structure.These generalized systems can be identified unambiguously by specifying the underlying loop algebra together with an ordered pair of integers (n, m). It turns out that different systems associated with the same underlying loop algebra but with different pairs of integers (n1, m1 ) and (n2, m2) with n2 ＜ n1 and m2 ＜ m1 can be related by a nested Hamiltonian reduction procedure. For all nontrivial generalizations, the extra coordinates besides the standard Toda variables are Poisson non-commute, and when either n or m ≥ 3, the Poisson structure for the extra coordinate variables becomes some Lie algebra (i.e. the extra variables appear linearly on the right-hand side of the Poisson brackets). In the quantum case, such generalizations will become systems with noncommutative variables without spoiling the integrability.