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Cluster integrable systems

Mikhail Bershtein

Abstract

In these lecture notes, we give an introduction to cluster integrable systems. The topics include relativistic Toda systems, moduli spaces of framed local systems, Goncharov-Kenyon integrable systems, and quantization.

Cluster integrable systems

Abstract

In these lecture notes, we give an introduction to cluster integrable systems. The topics include relativistic Toda systems, moduli spaces of framed local systems, Goncharov-Kenyon integrable systems, and quantization.

Paper Structure

This paper contains 29 sections, 25 theorems, 134 equations, 44 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Poisson manifolds and classical integrable systems
  4. Poisson manifolds
  5. Symplectic leaves
  6. Classical Integrable systems
  7. Poisson--Lie groups
  8. Poisson--Lie groups
  9. Double Bruhat cells
  10. Factorization schemes
  11. Cluster varieties
  12. Seeds
  13. Mutations
  14. Cluster variety
  15. Clusters and relativistic Toda system.
  16. ...and 14 more sections

Key Result

Theorem 3.5

The group $G$ has the following decompositions

Figures (44)

  • Figure 4.1: On the left matrix $\epsilon$, on the right the corresponding quiver
  • Figure 4.2: Example of mutation in the vertex 2
  • Figure 4.3: Quivers corresponding to Examples \ref{['Ex: ss']} and \ref{['Ex: s123']}
  • Figure 4.4: Pentagon for $A_2$ quiver
  • Figure 5.1: Quivers $\mathcal{Q}_{\bar{2}}$ and $\mathcal{Q}_{2}$ for $GL_5$
  • ...and 39 more figures

Theorems & Definitions (94)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3: Constant bracket
  • Example 2.4
  • Example 2.5: Symplectic manifolds
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • Example 2.10
  • ...and 84 more