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Integrable systems and cluster algebras

Michael Gekhtman, Anton Izosimov

Abstract

We review several constructions of integrable systems with an underlying cluster algebra structure, in particular the Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on perfect networks and the Goncharov-Kenyon approach based on the dimer model. We also discuss results of Galashin and Pylyavskyy on integrability of T-systems.

Integrable systems and cluster algebras

Abstract

We review several constructions of integrable systems with an underlying cluster algebra structure, in particular the Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on perfect networks and the Goncharov-Kenyon approach based on the dimer model. We also discuss results of Galashin and Pylyavskyy on integrability of T-systems.
Paper Structure (48 sections, 6 theorems, 13 equations, 14 figures)

This paper contains 48 sections, 6 theorems, 13 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Poisson manifolds and integrable systems
  4. Poisson manifolds, maps, and submanifolds.
  5. The rank of a Poisson structure, symplectic leaves, and Casimirs.
  6. Continuous integrable systems.
  7. Discrete integrable systems.
  8. Quivers, mutations, and cluster maps
  9. Quivers and quiver mutations.
  10. $X$-type cluster mutations.
  11. Canonical presymplectic structure.
  12. $Y$-type cluster mutations.
  13. Poisson property of $Y$-mutations.
  14. Cluster integrable systems from weighted directed networks
  15. Continuous systems
  16. ...and 33 more sections

Key Result

Theorem 4.1

GSV4 For any perfect network ${\mathcal{N}}$ on a cylinder, the above-defined Poisson structure on the space of edge weights induces the trigonometric R-matrix bracket on the space of boundary measurement matrices. In other words, the boundary measurement map is a Poisson map from the space ${\mathc

Figures (14)

  • Figure 1: Quiver mutation at vertex $1$.
  • Figure 2: $X$-type mutation at vertex $x_1$.
  • Figure 3: Quiver mutation at vertex $1$.
  • Figure 4: A perfect network ${\mathcal{N}}$ on a cylinder and its boundary measurement matrix.
  • Figure 5: A perfect network ${\mathcal{N}}$ on a cylinder (grey) and its dual quiver $\mathcal{Q}_{\mathcal{N}}$ (black). Dotted arrows are half-edges.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Example 3.1
  • Example 3.2
  • Theorem 4.1
  • Corollary 4.2
  • Remark 4.3
  • Example 5.1
  • Theorem 5.2
  • Remark 5.3: cf. Remark \ref{['pushforward']}
  • Theorem 6.1
  • Theorem 6.2
  • ...and 2 more