We define an infinite chain of subcategories of the partition category by introducing the left-height ($l$) of a partition. For the Brauer case, the chain starts with the Temperley-Lieb ($l=-1$) and ends with the Brauer ($l=infty$) category. The End
sets are algebras, i.e., an infinite tower thereof for each $l$, whose representation theory is studied in the paper.
For each natural number n greater than 1, we define an algebra satisfying many properties that one might expect to hold for a Brauer algebra of type Cn. The monomials of this algebra correspond to scalar multiples of symmetric Brauer diagrams on 2n s
trands. The algebra is shown to be free of rank the number of such diagrams and cellular, in the sense of Graham and Lehrer.
The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory.
In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over ground rings with parameters satisfying so-called admissibility conditions. These conditions are necessary in order for these results to hold and originally arise from the representation theory of B_2^k, which is analysed by the authors in a previous paper. Furthermore, we obtain a geometric realisation of B_n^k as a cyclotomic version of the Kauffman tangle algebra, in terms of affine n-tangles in the solid torus, and produce explicit bases that may be described both algebraically and diagrammatically. The admissibility conditions are the most general offered in the literature for which these results hold; they are necessary and sufficient for all results for general n.
----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced
by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.
