On geometry of 2-nondegenerate CR structures of hypersurface type and flag structures on leaf spaces of Levi foliations

Sykes, David; Zelenko, Igor

doi:10.1016/j.aim.2022.108850

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.

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