Generalized Toda mechanics associated with classical Lie algebras and their reductions
Abstract
For any classical Lie algebra
g
, we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers
(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=
B
r
,
C
r
,
D
r
with
m,n≤3
are also given. For all
m,n
, it is shown that the dynamics of the
(m,n−1)
- and the
(m−1,n)
-Toda chains 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
)
Sym
-Toda systems, which are also integrable. In the quantum case, all
(m,n)
-Toda systems with
m>1
or
n>1
describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all
(m,n)
-Toda systems survive after quantization.