Belavin–Drinfeld classification conjecture for compatible cluster structures on simple Lie groups

Let ${\mathcal G}$ be a simple complex Lie group. A Belavin–Drinfeld triple is a triple $T=(\Gamma_1,\Gamma_2,\gamma)$, and write $k_T=|\Delta\setminus\Gamma_1|$. The associated torus is $\mathcal H_T=\exp\mathfrak h_T\subset\mathcal G$. The Belavin–Drinfeld cluster-structure conjecture. For any Belavin–Drinfeld triple $T=(\Gamma_1,\Gamma_2,\gamma)$ there exists a cluster structure ${\mathcal C}_T$ on ${\mathcal G}$ such that (i) the number of stable variables is $2k_T$, and the corresponding extended exchange matrix has full rank; (ii) ${\mathcal C}_T$ is regular, and the corresponding upper cluster algebra $\overline{{\mathcal A}}_{\mathbb C}({\mathcal C}_T)$ is naturally isomorphic to ${\mathcal O}({\mathcal G})$; (iii) the global toric action of $(\mathbb C^*)^{2k_T}$ on $\overline{{\mathcal A}}_{\mathbb C}({\mathcal C}_T)$ is generated by the action of $\mathcal H_T\times\mathcal H_T$ on ${\mathcal G}$ given by $(H_1,H_2)(X)=H_1XH_2$; (iv) for any solution of the classical Yang–Baxter equation belonging to the Belavin–Drinfeld class specified by $T$, the corresponding Sklyanin bracket is compatible with ${\mathcal C}_T$; and (v) a Poisson–Lie bracket on ${\mathcal G}$ is compatible with ${\mathcal C}_T$ only if it is a scalar multiple of the Sklyanin bracket associated with a solution of the classical Yang–Baxter equation belonging to the Belavin–Drinfeld class specified by $T$. This conjecture seeks a classification of regular cluster structures on ${\mathcal G}$ parallel to the Belavin–Drinfeld classification. The paper presents supporting examples, including the standard Poisson–Lie structure for every simple complex Lie group and the full Belavin–Drinfeld classification for $SL_n$ with $n=2,3,4$; the general assertion remains open.