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Title:  The Cyclotomic BirmanMurakamiWenzl Algebras 
Authors:  Yu, Shona Huimin 
Keywords:  BMW algebras BirmanMurakamiWenzl algebras ArikiKoike algebras cyclotomic Hecke algebras type B tangles cellular 
Issue Date:  Oct2008 
Publisher:  School of Mathematics and Statistics 
Abstract:  This thesis presents a study of the cyclotomic BMW algebras, introduced by HaringOldenburg as a generalization of the BMW (BirmanMurakamiWenzl) algebras related to the cyclotomic Hecke algebras of type G(k,1,n) (also known as ArikiKoike algebras) and type B knot theory involving affine/cylindrical tangles. The motivation behind the definition of the BMW algebras may be traced back to an important problem in knot theory; namely, that of classifying knots (and links) up to isotopy. The algebraic definition of the BMW algebras uses generators and relations originally inspired by the Kauffman link invariant. They are intimately connected with the Artin braid group of type A, IwahoriHecke algebras of type A, and with many diagram algebras, such as the Brauer and TemperleyLieb algebras. Geometrically, the BMW algebra is isomorphic to the Kauffman Tangle algebra. The representations and the cellularity of the BMW algebra have now been extensively studied in the literature. These algebras also feature in the theory of quantum groups, statistical mechanics, and topological quantum field theory. In view of these relationships between the BMW algebras and several objects of "type A", several authors have since naturally generalized the BMW algberas for other types of Artin groups. Motivated by knot theory associated with the Artin braid group of type B, HaringOldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the ArikiKoike algebra h_{n,k} is a quotient of B_n^k, in the same way the IwahoriHecke algebra of type A is a quotient of the BMW algebra. In this thesis, we investigate the structure of these algebras and show they have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. In particular, they are shown to be Rfree of rank k^n (2n1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and ArikiKoike algebras, one must impose extra socalled "admissibility conditions" on the parameters of the ground ring in order for these results to hold. This is due to potential torsion caused by the polynomial relation of order k imposed on one of the generators of B_n^k. It turns out that the representation theory of B_2^k is crucial in determining these conditions precisely. The representation theory of B_2^k is analysed in detail in a joint preprint with Wilcox in [45] (http://arxiv.org/abs/math/0611518). The admissibility conditions and a universal ground ring with admissible parameters are given explicitly in Chapter 3. The admissibility conditions are also closely related to the existence of a nondegenerate Markov trace function of B_n^k which is then used together with the cyclotomic Brauer algebras in the linear independency arguments contained in Chapter 4. Furthermore, in Chapter 5, we prove the cyclotomic BMW algebras are cellular, in the sense of Graham and Lehrer. The proof uses the cellularity of the ArikiKoike algebras (GrahamLehrer [16] and DipperJamesMathas [8]) and an appropriate "lifting" of a cellular basis of the ArikiKoike algebras into B_n^k, which is compatible with a certain antiinvolution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by MortonWasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and HauschildMosley also attempt similar arguments to establish the freeness and diagram algebra results mentioned above. However, they withdrew their preprints ([14] and [15]), due to issues with their generic ground ring crucial to their linear independence arguments. A similar strategy to that proposed in [14], together with different trace maps and the study of rings with admissible parameters in Chapter 3, is used in establishing linear independency of our basis in Chapter 4. Since the submission of this thesis, new versions of these preprints have been released in which Goodman and HauschildMosley use alternative topological and Jones basic construction theory type arguments to establish freeness of B_n^k and an isomorphism with the cyclotomic Kauffman Tangle algebra. However, they require their ground rings to be an integral domain with parameters satisfying the (slightly stronger) admissibility conditions introduced by Wilcox and the author in [45]. Also, under these conditions, Goodman has obtained cellularity results. Rui and Xu have also obtained freeness and cellularity results when k is odd, and later Rui and Si for general k, under the assumption that \delta is invertible and using another stronger condition called "uadmissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic NazarovWenzl algebras and involve the construction of seminormal representations; their preprints have recently been released on the arXiv. It should also be noted there are slight differences between the definitions of cyclotomic BMW algebras and ground rings used, as explained partly above. Furthermore, Goodman and RuiSiXu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the antiinvolution axiom of the original GrahamLehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007. 
Description:  Doctor of Philosophy 
URI:  http://hdl.handle.net/2123/3560 
Rights and Permissions:  The author retains copyright of this thesis. 
Type of Work:  PhD Doctorate 
Appears in Collections:  Sydney Digital Theses (Open Access) 
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