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Journal of Lie Theory 21 (2011), No. 4, 929--960
Copyright Heldermann Verlag 2011

The Integrability of the Periodic Full Kostant-Toda Lattice on a Simple Lie Algebra

Khaoula Ben Abdeljelil
Laboratoire de Mathématiques, Route de Chartres, B. P. 6759, 45067 Orléans, France

We define the periodic Full Kostant-Toda lattice on every simple Lie algebra, and show its Liouville integrability. More precisely we show that this lattice is given by a Hamiltonian vector field, associated to a Poisson bracket which results from an R-matrix. We construct a large family of constants of motion which we use to prove the Liouville integrability of the system with the help of several results on simple Lie algebras, R-matrices, invariant functions and root systems.

Keywords: Periodic Full Kostant-Toda lattice, integrable system, R-matrix, simple Lie algebra.

MSC: 17B20,17B80,53D17

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