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We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$and $Ma{t}_n$that are compatible with Poisson–Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on a semisimple complex group $\mathcal{G}$corresponds to a cluster structure in $\mathcal{O}\left(\mathcal{G}\right)$. Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of $A_n$type, which includes all the previously known examples. Namely, we subdivide all possible $A_n$type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson–Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $S{L}_n$compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $S{L}_n$equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications. 

As is well known, cluster transformations in cluster structures of geometric type are often modeled on determinant identities, such as short Plücker relations, Desnanot– Jacobi identities, and their generalizations. We present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in GL_n compatible with a certain subclass of Belavin–Drinfeld Poisson–Lie brackets, in the Drinfeld double of GL_n, and in spaces of periodic difference operators. 

We continue the study of multiple cluster structures in the rings of regular functions on $$GL_n$$, $$SL_n$$ and $$\operatorname{Mat}_n$$ that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson--Lie structures on a semisimple complex group $$\mathcal G$$ corresponds to a cluster structure in $$\mathcal O(\mathcal G)$$. Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of $$A_n$$ type, which includes all the previously known examples. Namely, we subdivide all possible $$A_n$$ type BD data into oriented and non-oriented kinds. In the oriented case, we single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on $$SL_n$$ compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of $$SL_n$$ equipped with two different Poisson-Lie brackets. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address this situation in future publications. 

We prove that the regular generalized cluster structure on the Drinfeld double of 𝐺𝐿𝑛 constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) is complete and compatible with the standard Poisson–Lie structure on the double. Moreover, we show that for 𝑛 = 4 this structure is distinct from a previously known regular generalized cluster structure on the Drinfeld double, even though they have the same compatible Poisson structure and the same collection of frozen variables. Further, we prove that the regular generalized cluster structure on band periodic matrices constructed in Gekhtman, Shapiro, and Vainshtein (Int. Math. Res. Notes, 2022, to appear, arXiv:1912.00453) possesses similar compatibility and completeness properties.