On the complete integrability of the periodic quantum Toda lattice

TL;DR

This paper proves the complete integrability of the periodic quantum Toda lattice for all extended Dynkin diagrams, extending previous results and enabling broader applications in quantum cohomology of affine flag manifolds.

Contribution

It establishes the complete integrability for all extended Dynkin diagrams, generalizing earlier classical and specific Lie type cases.

Findings

Proves complete integrability for all extended Dynkin diagrams.

Extends quantum cohomology results to all Lie types.

Bridges gaps in previous classical and $E_6$ cases.

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 $E_6$ 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.