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A generalized cluster structure on $GL_n$ via birational Poisson maps

Misha Gekhtman, Michael Shapiro, Alek Vainshtein

Abstract

In a recent work, we constructed a rational map from a simple Lie group $\mathcal G$ to itself that intertwines the standard Poisson--Lie structure on $\mathcal G$ with a Poisson homogeneous one defined by a pair of quasi-triangular solutions to the classical Yang--Baxter equation (CYBE) known as R-matrices. We also showed, in the case of $SL_n$, that if the combinatorial Belavin--Drinfeld data associated with these R-matrices satisfies certain aperiodicity conditions, the map is, in fact, birational and can be used to obtain an initial cluster for an exotic cluster structure on $SL_n$ via the pullback of Berenstein--Fomin--Zelevinsky cluster variables. The same strategy was later used by the first author and D.~Voloshyn to describe generalized cluster structures compatible with the Poisson dual of the Poisson--Lie bracket defined by a quasi-triangular R-matrix. In this paper we further promote the use of birational Poisson maps in constructing generalized cluster structures by applying it in the situation when the aperiodicity condition is not satisfied. To this end, we describe a generalized cluster structure on $GL_n$ compatible with the Poisson homogeneous bracket defined by two Cremmer--Gervais solutions to the CYBE related via conjugation by the longest element of the Weyl group. The key ingredient to our construction is a birational map that connects the bracket under consideration with two other Poisson brackets: the Poisson dual to Cremmer--Gervais Poisson--Lie bracket on $GL_{n-1}$ and the bracket on a certain space of complex rational functions of one variable closely related to cluster algebraic interpretation of Coxeter--Toda flows. New notions of a regular pullback of a seed and of an almost-cluster structure whose detailed description are given in a separate note also play an important role in our construction.

A generalized cluster structure on $GL_n$ via birational Poisson maps

Abstract

In a recent work, we constructed a rational map from a simple Lie group to itself that intertwines the standard Poisson--Lie structure on with a Poisson homogeneous one defined by a pair of quasi-triangular solutions to the classical Yang--Baxter equation (CYBE) known as R-matrices. We also showed, in the case of , that if the combinatorial Belavin--Drinfeld data associated with these R-matrices satisfies certain aperiodicity conditions, the map is, in fact, birational and can be used to obtain an initial cluster for an exotic cluster structure on via the pullback of Berenstein--Fomin--Zelevinsky cluster variables. The same strategy was later used by the first author and D.~Voloshyn to describe generalized cluster structures compatible with the Poisson dual of the Poisson--Lie bracket defined by a quasi-triangular R-matrix. In this paper we further promote the use of birational Poisson maps in constructing generalized cluster structures by applying it in the situation when the aperiodicity condition is not satisfied. To this end, we describe a generalized cluster structure on compatible with the Poisson homogeneous bracket defined by two Cremmer--Gervais solutions to the CYBE related via conjugation by the longest element of the Weyl group. The key ingredient to our construction is a birational map that connects the bracket under consideration with two other Poisson brackets: the Poisson dual to Cremmer--Gervais Poisson--Lie bracket on and the bracket on a certain space of complex rational functions of one variable closely related to cluster algebraic interpretation of Coxeter--Toda flows. New notions of a regular pullback of a seed and of an almost-cluster structure whose detailed description are given in a separate note also play an important role in our construction.
Paper Structure (35 sections, 46 theorems, 297 equations, 14 figures)

This paper contains 35 sections, 46 theorems, 297 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. Poisson--Lie groups
  4. Generalized cluster structures and compatible Poisson brackets
  5. Regular pullbacks of a seed and almost-cluster structures
  6. Main result
  7. Generalized cluster structure on $GL_n^\dag$ compatible with the Cremmer--Gervais bracket
  8. Cluster structure on rational functions
  9. Outline of the proof
  10. Map $\Psi$ and the proof of Theorem \ref{['twomaps']}
  11. Map $\Psi'$ and the proof of Theorem \ref{['twomaps']}(i)
  12. Map $\Psi"$ and the proof of Theorem \ref{['twomaps']}(ii)
  13. Proof of Theorem \ref{['twomaps']}(iii)
  14. Explicit expressions: the proof of Theorem \ref{['expressions']}
  15. Proof of Theorem \ref{['expressions']}(i)
  16. ...and 20 more sections

Key Result

Proposition 2.1

The $y$-variables for the seed $\Psi^*\Sigma$ coincide with the pullback by $\Psi$ of $y$-variables for the seed $\Sigma$.

Figures (14)

  • Figure 1: Quiver $Q_5^\dag$
  • Figure 2: Quiver $Q_5^T$
  • Figure 3: Quiver $\bar{Q}_5^T$
  • Figure 4: Quivers $Q_{n-1}^\dag$ (on the left) and $\bar{Q}_{n-1}^T$ (on the right)
  • Figure 5: Quivers $\Psi'^*Q_{n-1}^\dag$ (on the left) and $\Psi"^*\bar{Q}_{n-1}^T$ (on the right)
  • ...and 9 more figures

Theorems & Definitions (82)

  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 72 more