Thesis (M. Phil.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 46-47).

In this dissertation, we study certain types of linear mappings on triangular algebras. Triangular algebras are algebras whose elements can be written in the form of 2 x 2 matrices [special characters omitted]where a ∈ A, b ∈ B, m ∈ M and where A, B are algebras and M is a bimodule. Many widely studied algebras, such as upper triangular matrix algebras and nest algebras, can be viewed as triangular algebras. This dissertation is divided into five chapters. The first chapter is a general account of the basics of triangular algebras, including the unitization of nonunital triangular algebras and the structure of the centre of triangular algebras, as well as a brief introduction to some well-known examples of triangular algebras. In Chapter 2, we study the general structure of derivations on triangular algebras and obtain some results on the first cohomology groups of triangular algebras. The first cohomology group of an algebra is the quotient space of the space of all derivations over the space of all inner derivations, and it is always a main tool in the research of derivations. In addition, we consider the problem of automatic continuity of derivations in the last section of this chapter. In Chapter 3, we consider sufficient conditions on a triangular algebra so that every Lie derivation is a sum of a derivation and a linear map whose image lies in the centre of the triangular algebra. In Chapter 4, we consider sufficient conditions for every commuting map on a triangular algebra to be a sum of a map of the form x ↦ ax and a map whose image lies in the centre of the triangular algebra. In the final chapter, we are concerned with the automorphisms of triangular algebras. The study of automorphism is a most important way to understand the underlying structure of an algebra. We deduce some results on the Skolem-Noether groups, or the outer automorphism groups, of triangular algebras and apply those results to generalize some known results about automorphisms on a triangular matrix algebras. / Graduate

The notion of the 'classical' multilinear maps such as the symmetric and skew-symmetric maps, has the following generalization: given a vector-space V and a pair (H[subscript n],X[subscript n]) where is a subgroup of the symmetric group S[subscript n] and X[subscript n] is a character of H[subscript n], we consider multilinear maps from V[superscript n] (n-fold cartesian product of V ) into any other vector space, which are ‘symmetric with respect to (H[subscript n],X[subscript n])’, i.e., which have a certain symmetry in their values on permuted tuples of vectors, where the permutations are in H[subscript n]. Given a pair (H[subscript n],X[subscript n]) and a vector-space V , we can construct a space V[superscript (n)] over V through which the maps 'symmetric with respect to (H[subscript n],X[subscript n])’ linearize. The space V[superscript (n)] is usually defined abstractly by means of a certain universal mapping property and gives the tensor, symmetric and Grassmarm spaces for the 'classical' maps. Given a sequence of pairs [formula omitted]and the corresponding spaces V[superscript (n)], we let [formula omitted] (where V[superscript b]) is the ground field). In the classical cases, A has a natural multiplicative operation which makes A an algebra, i.e., the Tensor, Symmetric and Grassmann algebras. This presentation has been motivated by the attempt to generalize the construction of an algebra A to a wider family of 'admissible' sequences of pairs [formula omitted]. This consideration has led us to investigate permutation groups on the numbers 1,2,3,… which are closed under a certain 'shift' of the permutations, i.e., if [formula omitted] is a permutation, we define [formula omitted] and we call a permutation group H 'translation-invariant' if for every [formula omitted] is also in H . We begin our presentation by characterising the 'translation-invariant' groups. We show that the study of these (infinite) groups can be reduced to the study of certain finite groups. Then, we proceed to discuss the lattice of the translation-invariant groups. Finally, we show that a translation-invariant group H , together, with an appropriate character X of H , represents an equivalence class of 'admissible' sequences of pairs [formula omitted]. For a particular choice of representatives of the equivalence class, we can construct an algebra of 'symmetry classes of tensors' which generalizes the Tensor, Symmetric and Grassmann algebras. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (D.A.)--Illinois State University, 1990. / Title from title page screen, viewed November 16, 2005. Dissertation Committee: Stephen Friedberg (chair), John Dossey, George Kidder, Michael Plantholt, Robert Ritt. Includes bibliographical references (leaves 22-23) and abstract. Also available in print.

Lau Yuk Hong. / Bibliography: leaves 63-64 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

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.

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.

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 14-15).

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