Compatible cluster structures for Belavin–Drinfeld classes

Let ${\mathcal G}$ be a simple complex Lie group. For any Belavin–Drinfeld triple $T=(\Gamma_1,\Gamma_2,\gamma)$, let $k_T=|\Delta\setminus\Gamma_1|$, let $\mathcal H_T=\exp\mathfrak h_T\subset {\mathcal G}$ be the associated torus, and let a Sklyanin bracket mean the Poisson–Lie bracket corresponding to a solution of the classical Yang–Baxter equation. A cluster structure $({\mathcal C}_T,\varphi_T)$ has stable variables and an extended exchange matrix, and its upper cluster algebra is denoted $\overline{{\mathcal A}}_{\mathbb C}({\mathcal C}_T)$.

Belavin–Drinfeld cluster-structure conjecture. For any Belavin–Drinfeld triple $T=(\Gamma_1,\Gamma_2,\gamma)$ there exists a cluster structure $({\mathcal C}_T,\varphi_T)$ on ${\mathcal G}$ such that:

1. the number of stable variables is $2k_T$, and the corresponding extended exchange matrix has full rank;
2. $({\mathcal C}_T,\varphi_T)$ is regular, and $\overline{{\mathcal A}}_{\mathbb C}({\mathcal C}_T)$ is naturally isomorphic to ${\mathcal O}({\mathcal G})$;
3. the global toric action of $(\mathbb C^*)^{2k_T}$ on $\mathbb C({\mathcal G})$ is generated by the action of $\mathcal H_T\times\mathcal H_T$ given by $(H_1,H_2)(X)=H_1XH_2$;
4. for any solution of the classical Yang–Baxter equation in the Belavin–Drinfeld class specified by $T$, the corresponding Sklyanin bracket is compatible with ${\mathcal C}_T$; and
5. 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 in the Belavin–Drinfeld class specified by $T$.

This precise conjecture refines the proposed parallel classification by assigning a regular cluster structure to each Belavin–Drinfeld triple and characterizing the compatible Poisson–Lie brackets. Its conclusions connect the upper cluster algebra with the coordinate ring of ${\mathcal G}$ and relate cluster toric actions to the tori determined by the triple.