Gekhtman–Shapiro–Vainshtein conjecture on cluster structures for Belavin–Drinfeld triples

Let $\mathcal{G}$ be a simple complex Lie group, let $T=(\Gamma_{1},\Gamma_{2},\gamma)$ be a Belavin–Drinfeld triple, and let $k_T$ and $\mathcal{H}_T$ be the quantities defined by the triple, with

$$\mathcal{H}_{T}=\exp\mathfrak{h}_{T},\qquad \mathfrak{h}_{T}=\{h\in\mathfrak{h}:\alpha(h)=\beta(h)\text{ if }\alpha\prec\beta\}.$$

A cluster structure $\mathcal{C}_T$ on $\mathcal{G}$ is expected to have $2k_T$ stable variables and a full-rank extended exchange matrix. Gekhtman–Shapiro–Vainshtein conjecture. For every Belavin–Drinfeld triple $T$, there exists a cluster structure $\mathcal{C}_T$ on $\mathcal{G}$ satisfying all of the following: it is regular; its upper cluster algebra $\overline{\mathcal{A}}_{\mathbb{C}}(\mathcal{C}_T)$ is naturally isomorphic to $\mathcal{O}(\mathcal{G})$; the global toric action of $(\mathbb{C}^{*})^{2k_T}$ on $\mathbb{C}(\mathcal{G})$ is generated by the action of $\mathcal{H}_T\otimes\mathcal{H}_T$ given by $(H_1,H_2)(X)=H_1XH_2$; every solution of the classical Yang–Baxter equation in the Belavin–Drinfeld class specified by $T$ has a compatible Sklyanin bracket; and a Poisson–Lie bracket on $\mathcal{G}$ is compatible with $\mathcal{C}_T$ only if it is a scalar multiple of such a Sklyanin bracket. This conjecture proposes a correspondence between Belavin–Drinfeld classes and cluster structures, including regularity, identification of the upper cluster algebra with the coordinate ring, and compatibility with the associated Poisson brackets; the stated existence and classification assertions are not resolved by the supplied text.