Multiple Generalized Cluster Structures on D(GLn)

We produce a large class of generalized cluster structures on the Drinfeld doubles of GLn and SLn compatible with a large class of Poisson brackets given by Belavin-Drinfeld classification. This work is a part of the grand research project led by Misha Gekhtman, Misha Shapiro and Alek Vainshtein, who aim at proving the following conjecture: for any given simple Poisson-Lie group endowed with a Poisson bracket from the Belavin-Drinfeld classification, there exists a compatible generalized cluster structure. The program naturally extends to Poisson duals and Drinfeld doubles of simple Poisson-Lie groups.