Some Representation Theory Of The Dilute Blob Algebra

The main objective of this thesis is to define a new class of multi-parameter algebras, called the dilute blob algebra dbn(p, q, r, s), which is a generalization of the Motzkin algebra. After we define basis diagrams of the dilute blob algebra, we give generators for the dilute blob algebra. A bijection between basis diagrams of the dilute blob algebra and basis diagrams of the left-right symmetric Motzkin algebra is also studied. We prove that the dilute blob algebra is cellular in the sense of Graham and Lehrer and construct the left cell modules. We then compute the dimension of these cell modules and the dimension of a dilute blob algebra. We define an inner product on these cell modules. Then we prove that the cell modules are cyclic. Moreover, we study the Gram matrix to determine when the cell module with n-1 propagating lines is simple. We also prove that the cell modules are generically simple over the complex field, thus the dilute blob algebra is generically semisimple over the complex field. We give a necessary and sufficient condition for a dilute blob algebra to be quasi-hereditary. Explicit restriction rules for the cell modules are given and we find the Bratelli diagram for n less or equal than 4. We also study induction of the cell modules