Cluster Structures on the Affine Space of Rectangular Matrices

It has been conjectured that there exists a classification of regular cluster structures on simple complex Lie groups that is parallel to the Belavin-Drinfeld classification of Poisson-Lie Brackets. On The general linear group one of these cluster structures corresponds to the Cremmer-Gervais Poisson bracket. This dissertation is dedicated to the study of a natural analogue of this cluster structure in the rings of regular functions on affine spaces of matrices. An outline is given for the construction of an initial cluster for a cluster structure on the space of rectangular n by m matrices that is compatible with the restriction of the Cremmer-Gervais Poisson bracket on the general linear group. Combinatorial conditions are given for when there exists a sequence of cluster mutations relating such cluster structures on the space of matrices of size n by m and the space of matrices of size (n-1) by m. This aforementioned sequence of mutations is an important ingredient in showing that the corresponding upper cluster algebra is isomorphic to the ring of regular functions on the space of matrices of size n by m.