The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, and talk about how quaternion algebras are important in Brauer groups by describing a theorem proved by Merkurjev in 1981.

The Yang-Baxter equation appear in various situations in physics and mathematics. For example it arises as a consistency condition in integrable models. The reflection equation (boundary Yang-Baxter equation) is a generalization of the Yang-Baxter equation to systems with a boundary. A further generalization to systems with defects which admits both reflection and transmission can be made, which results in reflection-transmission Yang-Baxter equations.In this thesis the Yang-Baxter equation and the reflection equation are presented. Representations of the Temperley-Lieb algebra and the blob algebra are used to construct matrices which solve the respective equations. For the reflection-transmission Yang-Baxter equations, steps toward a solution are taken by using a similar approach as for the first two cases, namely by finding an algebra whose representations can be used to construct matrices which solve the equations.

The Drinfel’d Lagrangian Grassmannian compactifies the space of algebraic maps of fixed degree from the projective line into the Lagrangian Grassmannian. It has a natural projective embedding arising from the highest weight embedding of the ordinary Lagrangian Grassmannian, and one may study its defining ideal in this embedding.The Drinfel’d Lagrangian Grassmannian is singular. However, a concrete description of generators for the defining ideal of the Schubert subvarieties of the Drinfel’d Lagrangian Grassmannian would implythat the singularities are modest. I prove that the defining ideal of any Schubert subvariety is generated by polynomials which give a straightening law on an ordered set. Using this fact, I show that any such subvariety is Cohen-Macaulay and Koszul. These results represent a partial extension of standard monomial theory to the Drinfel’d Lagrangian Grassmannian.

Thesis (M.S.)--Rutgers University, 2009. / "Graduate Program in Physics and Astronomy." Includes bibliographical references (p. 17).

Thesis (Ph.D.)--York University, 2008. Graduate Programme in Mathematics and Statistics. / Typescript. Includes bibliographical references (leaves 275-278). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:NR45986

Thesis--M. Phil., University of Hong Kong, 1980.

Not available / text

published_or_final_version / Mathematics / Master / Master of Philosophy

We find the generating function for group tensors contained in the enveloping algebra of each simple compact group of rank three or less. The generating function depends on dummy variables which carry, as exponents, the degrees and representation labels of the tensors; it suggests an integrity basis, a finite number of elementary tensors, in terms of which all can be expressed as stretched tensor products. We show how the generating functions for tensors in the enveloping algebra of SO(5) and SU(3) reduce when the tensors are acting on the basis of representations for which one of the Cartan labels vanish. The missing label problem in the reduction SO(5) (R-HOOK) SO(3) restricted to SO(5) representations of the type (0,(nu)) is considered; the eigenvalues and eigenvectors of a missing label operator are found up to (including) representation (0,12).