Loop algebras, gauge invariants and a new completely integrable system
Abstract
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 finitedimensional Lie algebra. This paper shows that the analogy is not perfect. A discrepancy arises because the natural generalization of the structure theory of finitedimensional simple Lie algebras is not the structure theory of loop algebras but the structure theory of affine KacMoody 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 Lietheoretic data. The commuting functions are precisely the gaugeinvariant functions one obtains by viewing elements of the loop algebra as connections on a bundle over $S^1$.
 Publication:

arXiv eprints
 Pub Date:
 June 1996
 DOI:
 10.48550/arXiv.solvint/9606008
 arXiv:
 arXiv:solvint/9606008
 Bibcode:
 1996solv.int..6008Q
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems