On the complete integrability of the periodic quantum Toda lattice

2016/04/29 by Augustin-Liviu Mare, Mare, Augustin-Liviu
Mathematics · #17B35 #37K10 #Dynamical Systems (math.DS) #FOS: Mathematics #Rings and Algebras (math.RA) #math.DS #math.RA #msc:17B35 #msc:37K10

paper · pdf · doi:10.48550/arxiv.1604.08726

19 pages; 2 figures. Final version, to appear in Forum Mathematicum

arxiv created 2016/11/30 · arxiv updated 2016/12/02

Abstract

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 E₆ by Goodman and Wallach at the beginning of the 1980's. 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.

Related

• Quantum cohomology and the periodic Toda lattice
• Loop algebras, gauge invariants and a new completely integrable system
• Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture
• Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups
• Quantum cohomology of the infinite dimensional generalized flag manifolds