On the complete integrability of the periodic quantum Toda lattice. (English) Zbl 1379.37105
Summary: We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and \(E_{6}\) by R. Goodman and N. R. Wallach [Commun. Math. Phys. 83, 355–386 (1982; Zbl 0503.22013); Commun. Math. Phys. 105, 473–509 (1986; Zbl 0616.22010)] at the beginning of the 1980s. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.
MSC:
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 81Q80 | Special quantum systems, such as solvable systems |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 17B35 | Universal enveloping (super)algebras |
| 17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
Keywords:
periodic quantum Toda lattice; complete integrability; Kac-Moody Lie algebra; Dynkin diagramReferences:
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