A Plethora of Cluster Structures on 𝐺𝐿_{𝑛}

Abstract

We continue the study of multiple cluster structures in the rings of regular functions on G L n GL_n , S L n SL_n and M a t n 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 G \mathcal {G} corresponds to a cluster structure in O ( G ) \mathcal {O}(\mathcal {G}) . Here we prove this conjecture for a large subset of Belavin–Drinfeld (BD) data of A n A_n type, which includes all the previously known examples. Namely, we subdivide all possible A n 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 SL_n compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of S L n SL_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.