Group theory approach to the \(\tau\)-function and its quantization
This is a review of generalizations of the \(\tau\)-function and integrable hierarchies and of their group theory interpretations, which admits an immediate quantization procedure. Different group theory structures related to the integrable system, as well as their quantum deformations, are discussed. Contents: 1. Introduction. 2. Fermionic representations for integrable hierarchies: General fermionic description of the standard integrable hierarchies, Bilinear identities for the classical 2DTL hierarchy, Fermionic representations for the forced hierarchies. 3. Group theory structures of the Toda molecule. 4. Generalized \(\tau\)-functions and bilinear identities: \(\tau\)-function and bilinear identities, Example: the quantum group \(\text{SL}_q(2)\), Universal \({\mathbf T}\)-operator (a group element), \(\tau\)-functions and representations of the algebra of functions, Difference KOS hierarchy. 5. Quantum and classical KP hierarchies with different evolutions: Structure of fundamental representations, Group element parametrization in fundamental representations, Classical \((q=1)\) KP/Toda hierarchy in different parametrizations, Quantum case \((q\neq 1)\). 6. Wave function and \(S\)-matrix in quantum Liouville (Toda) theory: Classical Toda system as a Hamiltonian reduction, Liouville quantum mechanis: general solution procedure, Liouville quantum mechanics: the group \(\text{SL}(2,\mathbb{R})\), Toda quantum mechanics. Appendix A: Forced hierarchies. Appendix B: Fundamental representations of the group \(\text{SL}_q(p)\). Appendix C: Explicit example: \(\tau_2\) for the group \(\text{SL}_q(p)\). Appendix D: Constructing the WF for the group \(\text{SL}(p)\).
scientific article; zbMATH DE number 1466087
scientific article