# Representation Theory Of Algebras Related To The Bubble Algebra

Hmaida, Mufida Mohamed A. (2016) Representation Theory Of Algebras Related To The Bubble Algebra. PhD thesis, University of Leeds.

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## Abstract

In this thesis we study several algebras which are related to the bubble algebra, including the bubble algebra itself. We introduce a new class of multi-parameter algebras, called the multi-colour partition algebra $P_{n,m} ( \breve{\delta} )$, which is a generalization of both the partition algebra and the bubble algebra. We also define the bubble algebra and the multi-colour symmetric groupoid algebra as sub-algebras of the algebra $P_{n,m} ( \breve{\delta} )$. We investigate the representation theory of the multi-colour symmetric groupoid algebra $\F S_{n,m}$. We show that $\F S_{n,m}$ is a cellular algebra and it is isomorphic to the generalized symmetric group algebra $\F \mathbb{Z}_m \wr S_n$ when $m$ is invertible and $\F$ is an algebraically closed field. We then prove that the algebra $P_{n,m} ( \breve{\delta} )$ is also a cellular algebra and define its cell modules. We are therefore able to classify all the irreducible modules of the algebra $P_{n,m} ( \breve{\delta} )$. We also study the semi-simplicity of the algebra $P_{n,m} ( \breve{\delta} )$ and the restriction rules of the cell modules to lower rank $n$ over the complex field. The main objective of this thesis is to solve some open problems in the representation theory of the bubble algebra $T_{n,m} ( \breve{\delta} )$. The algebra $T_{n,m} ( \breve{\delta} )$ is known to be cellular. We use many results on the representation theory of the Temperley-Lieb algebra to compute bases of the radicals of cell modules of the algebra $T_{n,m} ( \breve{\delta} )$ over an arbitrary field. We then restrict our attention to study representations of $T_{n,m} ( \breve{\delta} )$ over the complex field, and we determine the entire Loewy structure of cell modules of the algebra $T_{n,m} ( \breve{\delta} )$. In particular, the main theorem is Theorem 5.41.

Item Type: Thesis (PhD) Bubble algebra, cellular algebra, multi-colour symmetric groupoid and multi-colour partition algebra The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) miss M. M. A. Hmaida 23 Jan 2017 11:47 23 Jan 2017 11:47 http://etheses.whiterose.ac.uk/id/eprint/15987

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