LOOP ALGEBRAS, GAUGE INVARIANTS AND A NEW COMPLETELY INTEGRABLE SYSTEM

One fruitful motivating principle of much research on the family of in tegrable systems known as ''Toda lattices'' has been the heuristic ass umption that the periodic Toda lattice in an affine Lie algebra is dir ectly analogous to the nonperiodic Toda lattice in a finite-dimensiona l Lie algebra. This paper shows that the analogy is not perfect. A dis crepancy arises because the natural generalization of the structure th eory of finite-dimensional simple Lie algebras is not the structure th eory of loop algebras but the structure theory of affine Kac-Moody alg ebras. In this paper we use this natural generalization to construct t he natural analog of the nonperiodic Toda lattice. Surprisingly, the r esult is not the periodic Toda lattice but a new completely integrable system on the periodic Toda lattice phase space. This integrable syst em is prescribed purely in terms of Lie-theoretic data. The commuting functions are precisely the gauge-invariant functions one obtains by v iewing elements of the loop algebra as connections on a bundle over S- 1.