Generalized Cluster Structures Compatible with the Cremmer-Gervais Poisson Bracket on Rectangular Matrices

According to the conjecture given by Gekhtman-Shapiro-Vainshtein, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on a semisimple complex group G corresponds to a cluster structure in O(G). This dissertation continues the study of cluster structures in the rings of regular functions on the affine space of rectangular matrices that are compatible with Poisson structures. In particular, we construct a generalized cluster structure on Mat5×7 compatible with the restriction of the Cremmer-Gervais Poisson bracket on GL7. We also provide a detailed description of a conjectural generalized cluster structure on Matm×n compatible with the restriction of the Cremmer-Gervais Poisson bracket on GLn.