Quantum Integrable Daha Dim

---
name: quantum-integrable-daha-dim
description: "Quantum integrable systems from DAHA (Double Affine Hecke Algebra) and DIM (Ding-Iohara-Miki) algebra connections. Covers Macdonald polynomials, Koornwinder polynomials, Ruijsenaars systems, and type A vs type C system distinctions. Triggers: DAHA, DIM algebra, Macdonald polynomials, Koornwinder polynomials, Ruijsenaars system, quantum integrable systems, quantum toroidal algebra, type A system, type C system."
category: quantum-math
---

# Quantum Integrable Systems from DAHA and DIM Algebra

## Core Connections

### DIM Algebra → Integrable Systems
- DIM (quantum toroidal algebra of gl_1) connects to quantum many-particle integrable systems
- Ruijsenaars trigonometric system as the typical example
- Eigenfunctions: Noumi-Shiraishi power series, Macdonald polynomials, Baker-Akhiezer functions

### DAHA → DIM Correspondence
- Ruijsenaars Hamiltonians ↔ Cherednik DAHA Hamiltonians of type A
- DIM algebra ↔ spherical DAHA correspondence
- DAHA eigenfunctions = non-symmetric Macdonald polynomials

## Type A vs Type C Distinctions

### Type A
- Standard Ruijsenaars system
- Macdonald polynomials as eigenfunctions
- Non-symmetric Macdonald = DAHA eigenfunctions

### Type C∨C
- Koornwinder Hamiltonians
- Symmetric Koornwinder polynomials as eigenfunctions
- Non-symmetric Koornwinder polynomials as DAHA eigenfunctions
- Generated from spherical version of type C∨C DAHA

## Applications

1. Solving quantum integrable systems using DAHA/DIM algebra methods
2. Computing Macdonald and Koornwinder polynomials
3. Analyzing type A vs type C system differences
4. Twisted integrable systems construction