We investigate cofree coalgebras, and limits and colimits of coalgebras in some abelian monoidal categories of interest, such as bimodules over a ring, and modules and comodules over a bialgebra or Hopf algebra. We nd concrete generators for the categories of coalgebras in these monoidal categories, and explicitly construct cofree coalgebras, products and limits of coalgebras in each case. This answers an open question in [4] on the existence of a cofree coring, and constructs the cofree (co)module coalgebra on a B-(co)module, for a bialgebra B.

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For an inclusion N \subseteq M of finite von Neumann algebras, we study the group of normalizers N_M(B) = {u: uBu^* = B} and the von Neumann algebra it generates. In the first part of the dissertation, we focus on the special case in which N \subseteq M is an inclusion of separable II_1 factors. We show that N_M(B) imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of N_M(B)'', this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of N_M(B)'' in terms of a unique countable subgroup of N_M(B). We then apply these general techniques to obtain results for inclusions B \subseteq M arising from the crossed product, group von Neumann algebra, and tensor product constructions. Our work also leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N \subseteq M is a regular inclusion of II_1 factors, then N norms M: These new results and techniques develop further the study of normalizers of subfactors of II_1 factors. The second part of the dissertation is devoted to studying normalizers of maximal abelian self-adjoint subalgebras (masas) in nonseparable II_1 factors. We obtain a characterization of masas in separable II_1 subfactors of nonseparable II_1 factors, with a view toward computing cohomology groups. We prove that for a type II_1 factor N with a Cartan masa, the Hochschild cohomology groups H^n(N,N)=0, for all n \geq 1. This generalizes the result of Sinclair and Smith, who proved this for all N having separable predual.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 53). Also available in electronic version. Access restricted to campus users.

The algebra of matrices M with entries in an abelian von Neumann algebra is a C*-module. C*-modules were originally defined and studied by Kaplansky and we outline the foundations of the theory and particular properties of M. Furthermore, we prove a structure theorem for ultraweakly closed submodules of M, using techniques from the theory of type I finite von Neumann algebras. By analogy with the classical join in topology, the join for operator algebras A and B acting on Hilbert spaces H and K, respectively, was defined by Gilfeather and Smith. Assuming that K is finite dimensional, Gilfeather and Smith calculated the Hochschild cohomology groups of the join. We assume that M is the algebra of matrices with entries in a maximal abelian von Neumann algebra U, A is an operator algebra acting on a Hilbert space K, and B is an ultraweakly closed subalgebra of M containing U. In this new context, we redefine the join, generalize the calculations of Gilfeather and Smith, and calculate the cohomology groups of the join.

The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the nite general linear group GL(n, q); together with character tables of some of its subgroups. We study the structure of GL(n, q) and some of its important subgroups such as SL(n, q); UT(n, q); SUT(n, q); Z(GL(n, q)); Z(SL(n, q)); GL(n, q)0 ; SL(n, q)0 ; the Weyl group W and parabolic subgroups P : In addition, we also discuss the groups PGL(n, q); PSL(n, q) and the a ne group A (n, q); which are related to GL(n, q): The character tables of GL(2; q); SL(2; q); SUT(2; q) and UT(2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n, q) is given, where the theories of irreducible polynomials and partitions of i 2 f1; 2; ; ng form the atoms from where each conjugacy class of GL(n, q) is constructed. We give a special attention to some elements of GL(n, q); known as regular semisimple, where we count the number and orders of these elements. As an example we compute the conjugacy classes of GL(3; q): Characters of GL(n, q) appear in two series namely, principal and discrete series characters. The process of the parabolic induction is used to construct a large number of irreducible characters of GL(n, q) from characters of GL(n, q) for m < n: We study some particular characters such as Steinberg characters and cuspidal characters (characters of the discrete series). The latter ones are of particular interest since they form the atoms from where each character of GL(n, q) is constructed. These characters are parameterized in terms of the Galois orbits of non-decomposable characters of F q n: The values of the cuspidal characters on classes of GL(n, q) will be computed. We describe and list the full character table of GL(n, q): There exists a duality between the irreducible characters and conjugacy classes of GL(n, q); that is to each irreducible character, one can associate a conjugacy class of GL(n, q): Some aspects of this duality will be mentioned. / Thesis (M.Sc. (School of Mathematical Sciences)) - University of KwaZulu-Natal, Pietermaritzburg, 2008.

Thesis (Ph. D.)--University of Oregon, 2006. / Typescript. Includes vita and abstract. "In this dissertation I investigate Poisson structures on symmetric and exterior algebras of modules over complex reductive Lie algebras. I use the results to study the braided symmetric and exterior algebras"--P. 1. Includes bibliographical references (leaves 150-152). Also available for download via the World Wide Web; free to University of Oregon users.

The abstract Banach associative symmetrical *-algebras over C, so called C*- algebras, were introduced first in 1943 by Gelfand and Naimark24. In the present time the theory of C*-algebras has become a vast portion of functional analysis having connections and applications in almost all branches of modern mathematics and theoretical physics. From the 1940’s and the beginning of 1950’s there were numerous attempts made to extend the theory of C*-algebras to a category wider than Banach algebras. For example, in 1952, while working on the theory of locally-multiplicatively-convex algebras as projective limits of projective families of Banach algebras, Arens in the paper8 and Michael in the monograph48 independently for the first time studied projective limits of projective families of functional algebras in the commutative case and projective limits of projective families of operator algebras in the non-commutative case. In 1971 Inoue in the paper33 explicitly studied topological *-algebras which are topologically -isomorphic to projective limits of projective families of C*-algebras and obtained their basic properties. He as well suggested a name of locally C*-algebras for that category. For the present state of the theory of locally C*-algebras see the monograph of Fragoulopoulou. Also there were many attempts to extend the theory of C*-algebras to nonassociative algebras which are close in properties to associative algebras (in particular, to Jordan algebras). In fact, the real Jordan analogues of C*-algebras, so called JB-algebras, were first introduced in 1978 by Alfsen, Shultz and Størmer in1. One of the main results of the aforementioned paper stated that modulo factorization over a unique Jordan ideal each JB-algebra is isometrically isomorphic to a JC-algebra, i.e. an operator norm closed Jordan subalgebra of the Jordan algebra of all bounded self-adjoint operators with symmetric multiplication acting on a complex Hilbert space. Projective limits of Banach algebras have been studied sporadically by many authors since 1952, when they were first introduced by Arens8 and Michael48. Projective limits of complex C*-algebras were first mentioned by Arens. They have since been studied under various names by Wenjen, Sya Do-Shin, Brooks, Inoue, Schmüdgen, Fritzsche, Fragoulopoulou, Phillips, etc. We will follow Inoue33 in the usage of the name "locally C*-algebras" for these objects. At the same time, in parallel with the theory of complex C*-algebras, a theory of their real and Jordan analogues, namely real C*-algebras and JB-algebras, has been actively developed by various authors. In chapter 2 we present definitions and basic theorems on complex and real C*-algebras, JB-algebras and complex locally C*-algebras to be used further. In chapter 3 we define a real locally Hilbert space HR and an algebra of operators L(HR) (not bounded anymore) acting on HR. In chapter 4 we give new definitions and study several properties of locally C*- and locally JB-algebras. Then we show that a real locally C*-algebra (locally JBalgebra) is locally isometric to some closed subalgebra of L(HR). In chapter 5 we study complex and real Abelian locally C*-algebras. In chapter 6 we study universal enveloping algebras for locally JB-algebras. In chapter 7 we define and study dual space characterizations of real locally C* and locally JB-algebras. In chapter 8 we define barreled real locally C* and locally JB-algebras and study their representations as unbounded operators acting on dense subspaces of some Hilbert spaces. It is beneficial to extend the existing theory to the case of real and Jordan analogues of complex locally C*-algebras. The present thesis is devoted to study such analogues, which we call real locally C*- and locally JB-algebras. / Mathematics / D. Phil. (Mathematics)

vii, 41 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We generalize a theorem of Zhu relating the trace of certain vertex algebra representations and modular invariants to the arena of vertex super algebras. The theorem explains why the space of simple characters for the Neveu-Schwarz minimal models NS( p, q ) is modular invariant. It also expresses negative products in terms of positive products, which are easier to compute. As a consequence of the main theorem, the subleading coefficient of the singular vectors of NS( p, q ) is determined for p and q odd. An interesting family of q -series identities is established. These consequences established here generalize results of Milas in this field. / Adviser: Arkady Vaintrob

ix, 114 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / In this work we prove that the finite W -algebras associated to nilpotent elements in the symplectic or orthogonal Lie algebras whose Jordan blocks are all the same size are quotients of twisted Yangians. We use this to classify the finite dimensional irreducible representations of these finite W -algebras. / Committee in charge: Jonathan Brundan, Co-Chairperson, Mathematics; Victor Ostrik, Co-Chairperson, Mathematics; Arkady Berenstein, Member, Mathematics; Hal Sadofsky, Member, Mathematics; Christopher Wilson, Outside Member, Computer & Information Science