Completely Integrable Systems Associated with Classical Root Systems

• Main Author: Toshio Oshima
• Format: Article
• Language: English
• Published: National Academy of Science of Ukraine 2007-04-01
• Series: Symmetry, Integrability and Geometry: Methods and Applications
• Subjects: completely integrable systems; Calogero-Moser systems; Toda lattices with boundary conditions
• Online Access: http://www.emis.de/journals/SIGMA/2007/061/

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 $B_n$-invariants. We review conditions supporting the conjecture and give a new condition assuring it.