Plethora of cluster structures on $G{L}_n$

Abstract

We continue the study of multiple cluster structures in the rings of regular functions on $G{L}_n$, $S{L}_n$ and ${\mathrm{Mat}}_n$ 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 $S{L}_n$ 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. 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.

92 pages, 21 figures

References in corpus (1)

• Cluster algebra structures on Poisson nilpotent algebras