Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 061, 50 pages      math-ph/0502028
Contribution to the Vadim Kuznetsov Memorial Issue

Completely Integrable Systems Associated with Classical Root Systems

Toshio Oshima
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan

Received December 14, 2006, in final form March 19, 2007; Published online April 25, 2007

We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the Bn-invariants. We review conditions supporting the conjecture and give a new condition assuring it.

Key words: completely integrable systems; Calogero-Moser systems; Toda lattices with boundary conditions.

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