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Regular pullback of generalized cluster structures

Misha Gekhtman, Michael Shapiro, Alek Vainshtein

Abstract

We consider the problem of lifting a regular cluster structure on a quasi-affine variety to the ambient affine space and a similar problem of defining a regular pullback of a regular cluster structure under a dominant rational map between affine spaces. We provide sufficient conditions for the existence of the corresponding object, called an almost-cluster structure, study its combinatorics, compatible Poisson bracket and the corresponding upper cluster algebra.

Regular pullback of generalized cluster structures

Abstract

We consider the problem of lifting a regular cluster structure on a quasi-affine variety to the ambient affine space and a similar problem of defining a regular pullback of a regular cluster structure under a dominant rational map between affine spaces. We provide sufficient conditions for the existence of the corresponding object, called an almost-cluster structure, study its combinatorics, compatible Poisson bracket and the corresponding upper cluster algebra.
Paper Structure (10 sections, 10 theorems, 37 equations, 4 figures)

This paper contains 10 sections, 10 theorems, 37 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General question
  3. Sufficient condition for the existence of a coherent pullback
  4. Combinatorics of a coherent pullback
  5. The pullback of a seed
  6. Mutations of a coherent pullback
  7. The simplest case: $d_m=1$
  8. The case $d_m>1$
  9. Compatible pullback of a compatible Poisson bracket
  10. Regular pullback of the upper cluster algebra

Key Result

Theorem 3.1

A coherent pullback of a regular (generalized) cluster structure ${\mathcal{C}}$ on $V$ with the initial cluster $F=\{f_l\}_{l=1}^s$ exists if for any $k\in K$ the family $(\hat{{\tt F}}\cup{\tt G}\cup{\tt H})\setminus\{{\tt q}_{k}\}$ is algebraically independent on $\{{\tt q}_{k}=0\}\subset{\mathbb

Figures (4)

  • Figure 1: Quiver $Q_3$: a) initial quiver; b) quiver after mutation at 2
  • Figure 2: Quiver and its pullbacks: a) initial quiver $Q$; b) quiver $\widehat{Q}$; c) quiver $\Psi^*\widehat{Q}$
  • Figure 3: Mutations of $Q_3$ (left) and $\widehat{Q}_3$ (right)
  • Figure 4: Upper convex hulls $C_{\tt h}$ and support lines

Theorems & Definitions (29)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 19 more