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 $\mathcal A$ 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 $\mathcal A(\Sigma_1)$, $\mathcal A(\Sigma_2)$ corresponding to $H_1$ and $H_2$ such that $f$ is an isomorphism between $\mathcal A(\Sigma_1)$ and $\mathcal A(\Sigma_2)$. 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.