One fruitful motivating
principle of much research on the family of integrable systems known as “Toda
lattices” has been the heuristic assumption that the periodic Toda lattice in an
affine Lie algebra is directly analogous to the nonperiodic Toda lattice in a
finite-dimensional Lie algebra. This paper shows that the analogy is not perfect. A
discrepancy arises because the natural generalization of the structure theory of
finite-dimensional simple Lie algebras is not the structure theory of loop algebras but
the structure theory of affine Kac-Moody algebras. In this paper we use this
natural generalization to construct the natural analog of the nonperiodic Toda
lattice. Surprisingly, the result is not the periodic Toda lattice but a new
completely integrable system on the periodic Toda lattice phase space. This
integrable system is prescribed purely in terms of Lie-theoretic data. The
commuting functions are precisely the gauge-invariant functions one obtains
by viewing elements of the loop algebra as connections on a bundle over
S1.
