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by Wong Wah Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1996. / Includes bibliographical references (leaves 53-54). / Introduction --- p.2 / Chapter 1 --- Preliminaries --- p.3 / Chapter 1.1 --- A Summary on C*-algebras --- p.3 / Chapter 1.2 --- Hereditary C*-subalgebras --- p.5 / Chapter 1.3 --- C*-Inductive Limit --- p.7 / Chapter 1.4 --- Basic K-Theory of C*-algebras --- p.9 / Chapter 2 --- C*-algebras of Real Rank Zero --- p.14 / Chapter 2.1 --- Basic Properties of C*-algebras with Real Rank Zero --- p.14 / Chapter 2.2 --- Strongly Morita Equivalence and Extensions --- p.30 / Chapter 3 --- Simple C*-algebras --- p.42 / Chapter 3.1 --- Basic Properties --- p.42 / Chapter 3.2 --- Cuntz Algebras --- p.51 / Bibliography --- p.53

by Chau Man Pan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 74-75). / Abstracts in English and Chinese. / Chapter 1 --- Preliminaries --- p.6 / Chapter 1.1 --- General Theory of C* algebras --- p.6 / Chapter 1.2 --- Hilbert Modules and Induced representations --- p.11 / Chapter 1.3 --- Crossed Products --- p.15 / Chapter 2 --- Twisted crossed products --- p.18 / Chapter 2.1 --- Basic definition --- p.18 / Chapter 2.2 --- Iterated twisted crossed products --- p.24 / Chapter 3 --- Induced representations --- p.27 / Chapter 3.1 --- Construction of Imprimitivity bimodule --- p.27 / Chapter 3.2 --- Basic theory about induced representations --- p.32 / Chapter 4 --- Ideal Theory --- p.38 / Chapter 4.1 --- Induction and Restriction processes --- p.38 / Chapter 4.2 --- Sub-quotients of twisted crossed products --- p.48 / Chapter 5 --- Mackey Machine --- p.53 / Chapter 5.1 --- Quasi regular systems --- p.53 / Chapter 5.2 --- First Step of Mackey Machine --- p.57 / Chapter 5.3 --- Second step of Mackey Machine --- p.59 / Chapter 6 --- Abelian systems --- p.64 / Chapter 6.1 --- Dual spaces of Abelian systems --- p.64 / Chapter 7 --- Appendix --- p.69 / Chapter 7.1 --- Classical version of induced representation --- p.69

Chan, Wai-Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (leaves 52-53). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.iii / Introduction --- p.iv / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- C*-algebras --- p.1 / Chapter 1.2 --- Topological groups --- p.8 / Chapter 2 --- Property T for topological groups --- p.18 / Chapter 2.1 --- Definitions and some basic properties --- p.18 / Chapter 2.2 --- Hereditary properties --- p.23 / Chapter 2.3 --- A characterization for property T --- p.26 / Chapter 2.4 --- Examples --- p.32 / Chapter 3 --- Property T for C*-algebras --- p.34 / Chapter 3.1 --- Countable discrete groups and their group C*- algebras --- p.34 / Chapter 3.2 --- Property T and nuclearity --- p.46 / Bibliography --- p.52

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The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.

Doctor of Philosophy / This thesis presents a study of the cyclotomic BMW algebras, introduced by Haring-Oldenburg as a generalization of the BMW (Birman-Murakami-Wenzl) algebras related to 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. 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, Iwahori-Hecke algebras of type A, and with many diagram algebras, such as the Brauer and Temperley-Lieb 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, Haring-Oldenburg introduced the cyclotomic BMW algebras B_n^k as a generalization of the BMW algebras such that the Ariki-Koike algebra h_{n,k} is a quotient of B_n^k, in the same way the Iwahori-Hecke 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 R-free of rank k^n (2n-1)!! and bases that may be explicitly described both algebraically and diagrammatically in terms of cylindrical tangles are obtained. Unlike the BMW and Ariki-Koike algebras, one must impose extra so-called "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 non-degenerate 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 Ariki-Koike algebras (Graham-Lehrer [16] and Dipper-James-Mathas [8]) and an appropriate "lifting" of a cellular basis of the Ariki-Koike algebras into B_n^k, which is compatible with a certain anti-involution of B_n^k. When k = 1, the results in this thesis specialize to those previously established for the BMW algebras by Morton-Wasserman [30], Enyang [9], and Xi [47]. REMARKS: During the writing of this thesis, Goodman and Hauschild-Mosley 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 Hauschild-Mosley 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 "u-admissibility". The methods and arguments employed are strongly influenced by those used by Ariki, Mathas and Rui [3] for the cyclotomic Nazarov-Wenzl 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 Rui-Si-Xu use a weaker definition of cellularity, to bypass a problem discovered in their original proofs relating to the anti-involution axiom of the original Graham-Lehrer definition. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.

<p>In this work we investigate the structure of this type of Lie algebras over arbitrary fields F by constructing them from their Abelian ideal. To accomplish this, an algorithm is developed and as application a classification up to 7-dimensional Lie Algebras is given. We discuss a recent example of financial mathematics as well.</p>

Thesis (Ph. D.)--Oregon State University, 1970. / Typescript (photocopy). Includes bibliographical references (leaves 81-82). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.