Generalized Toda mechanics associated with classical Lie algebras and their reductions


For any classical Lie algebra gg, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers (m,n)(m,n). The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for g=Br,Cr,Dr\mathfrak{g}=B_{r},C_{r},D_{r} with m,n≀3m,n\leq3 are also given. For all m,nm,n, it is shown that the dynamics of the (m,nβˆ’1)(m,n-1)- and the (mβˆ’1,n)(m-1,n)-Toda chains are natural reductions of that of the (m,n)(m,n)-chain, and for m=nm=n, there is also a family of symmetrically reduced Toda systems, the (m,m)Sym(m,m)_{\mathrm{Sym}}-Toda systems, which are also integrable. In the quantum case, all (m,n)(m,n)-Toda systems with m>1m>1 or n>1n>1 describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all (m,n)(m,n)-Toda systems survive after quantization.Comment: 19 pages, bibte

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