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On Galois theory of cluster algebras: general and that from Riemann surfaces

Jinlei Dong, Fang Li

Abstract

One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois theory, we want to discuss the relations between cluster subalgebras of a cluster algebra and subgroups of its automorphism group and then set up the Galois-like method. In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra $\mathcal A$ to its cluster automorphism group, and introduce notions of Galois-like extensions and Galois extensions. A necessary condition for Galois extensions of a cluster algebra $\mathcal A$ is given, which is also a sufficient condition if $\mathcal A$ has a $\mathcal{D}$-stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups $H_1$ and $H_2$ of the automorphism group $\text{Aut}\mathcal A$ are conjugate to each other if and only if there exists $ f \in \text{Aut}\mathcal{A} $ and two Galois-like extension subalgebras $\mathcal A(Σ_1)$, $\mathcal A(Σ_2)$ corresponding to $H_1$ and $H_2$ such that $f$ is an isomorphism between $\mathcal A(Σ_1)$ and $\mathcal A(Σ_2)$. In the second part, as the answers of two conjectures proposed in the first part, for a cluster algebra from a feasible surface, we prove that Galois-like extension subalgebras corresponding to a subgroup of a cluster automorphism group have the same rank. Moreover, it is shown that there are order-preserving reverse Galois maps for these cluster algebras. We also give examples of $\mathcal{D}$-stable bases and some discussions on the Galois inverse problem in this part.

On Galois theory of cluster algebras: general and that from Riemann surfaces

Abstract

One of the key points in Galois theory via field extensions is to build up a correspondence between subfields of a field and subgroups of its automorphism group, so as to study fields via methods of groups. As an analogue of the Galois theory, we want to discuss the relations between cluster subalgebras of a cluster algebra and subgroups of its automorphism group and then set up the Galois-like method. In the first part, we build up a Galois map from a skew-symmetrizable cluster algebra to its cluster automorphism group, and introduce notions of Galois-like extensions and Galois extensions. A necessary condition for Galois extensions of a cluster algebra is given, which is also a sufficient condition if has a -stable basis or stable monomial basis with unique expression. Some properties for Galois-like extensions are discussed. It is shown that two subgroups and of the automorphism group are conjugate to each other if and only if there exists and two Galois-like extension subalgebras , corresponding to and such that is an isomorphism between and . In the second part, as the answers of two conjectures proposed in the first part, for a cluster algebra from a feasible surface, we prove that Galois-like extension subalgebras corresponding to a subgroup of a cluster automorphism group have the same rank. Moreover, it is shown that there are order-preserving reverse Galois maps for these cluster algebras. We also give examples of -stable bases and some discussions on the Galois inverse problem in this part.
Paper Structure (9 sections, 34 theorems, 75 equations, 6 figures)

This paper contains 9 sections, 34 theorems, 75 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on cluster algebras and Riemann oriented surfaces
  3. Cluster automorphism groups, extensions and $\mathcal{D}$-stable bases for cluster algebras
  4. Galois group, Galois map and cluster automorphism groups embedded into $S_{n} \rtimes \mathbb{Z}_{2}$
  5. Galois-like extensions, conjugate of subgroups and some conjectures
  6. Galois extensions, stable monomial bases and $\mathcal{D}$-stable bases for cluster algebras
  7. Examples of $\mathcal{D}$-stable bases and Galois theory of cluster algebras from surfaces
  8. Bangle, bracelet, and band bases as $\mathcal{D}$-stable bases
  9. Reverse Galois maps and the Galois inverse problem for cluster algebras from feasible surfaces

Key Result

Theorem 1

$H_{1}$ is conjugate to $H_{2}$ if and only if there exist $f \in \text{\em Aut}\mathcal{A}$, $\mathcal{A}(\Sigma_{1}) \in \mathcal{M}_{sub}^{H_{1}}$, $\mathcal{A}(\Sigma_{2}) \in \mathcal{M}_{sub}^{H_{2}}$ such that the restriction of $f$ is an isomorphism between $\mathcal{A}(\Sigma_{1})$ and $\ma

Figures (6)

  • Figure 1: A bangle $\text{Bang}_3 \zeta$(left), a bracelet $\text{Brac}_3 \zeta$(middle) and a band $\text{Band}_{3} \zeta$ (right).
  • Figure 2: The smoothing of $\zeta$ and $\alpha$
  • Figure 3: The local region from $\gamma$ to $\delta$
  • Figure 4: Case (1)
  • Figure 5: Case (2)
  • ...and 1 more figures

Theorems & Definitions (81)

  • Theorem 1: Theorem \ref{['galad']}
  • Theorem 2: Theorem \ref{['crit']}
  • Theorem 3: Theorem \ref{['c1']}
  • Theorem 4: Theorem \ref{['c2']}
  • Proposition 5: Proposition \ref{['prop-Galois-inverse-problem']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 71 more