Quantum and Classical Integrable Systems

Abstract

The key concept discussed in these lectures is the relation between the Hamiltonians of a quantum integrable system and the Casimir elements in the underlying hidden symmetry algebra. (In typical applications the latter is either the universal enveloping algebra of an affine Lie algebra, or its q-deformation.) A similar relation also holds in the classical case. We discuss different guises of this very important relation and its implication for the description of the spectrum and the eigenfunctions of the quantum system. Parallels between the classical and the quantum cases are thoroughly discussed.

59 pages, LaTeX2.09 with AMS symbols. Lectures at the CIMPA Winter School on Nonlinear Systems, Pondicherry, January 1996

References in corpus (6)

• Separation of Variables. New Trends.
• Gaudin Model, Bethe Ansatz and Critical Level
• Algebraic Aspects of Bethe-Ansatz
• Abelian Current Algebra and the Virasoro Algebra on the Lattice
• Affine Algebras, Langlands Duality and Bethe Ansatz
• Classical and Quantum Integrable Systems in $\wt{\gr{gl}}(2)^{+*}$ and Separation of Variables

Cited by in corpus (11)

• Integrable systems from the classical reflection equation
• Trigonometric osp(1|2) Gaudin model
• Algebraic Bethe ansatz for the XXX chain with triangular boundaries and Gaudin model
• sl_2 Gaudin model with Jordanian twist
• Trigonometric sl(2) Gaudin model with boundary terms
• Algebraic Bethe ansatz for the sl(2) Gaudin model with boundary
• Algebraic Bethe ansatz for the XXZ Heisenberg spin chain with triangular boundaries and the corresponding Gaudin model
• Bäcklund Transformations as exact integrable time-discretizations for the trigonometric Gaudin model
• Algebraic Bethe Ansatz for deformed Gaudin model
• Quantization of classical integrable systems. Part I: quasi-integrable quantum systems
• Jordanian deformation of the open sl(2) Gaudin model