Date of Award
Doctor of Philosophy (PhD)
An attempt was made to make this a self-contained reading. The first three chapters are intended to provide the necessary background. Chapter one develops the tools needed from Galois Cohomology. Chapter two is a brief description of involutions, and in chapter three we define the notion of (linear) algebraic group, we give some examples and discuss some of their properties. In chapter four, we discuss some variants of the classical Skolem-Noether theorem, requiring only that the subalgebra have a unique faithful representation of full degree over a separable closure. We verify that we can extend every isomorphism to the whole algebra by means of inner automorphisms, just as in the classical case. Examples of algebras that satisfy this condition are simple algebras and commutative Frobenius algebras. In chapter five, we attach involutions to our algebras. We show that Skolem-Noether type results hold over a separable closure and we discuss some descent problems. Chapter six is a study of k-conjugacy classes of maximal k-tori, the main goal of this dissertation. We are able to give explicit descriptions of k-conjugacy classes in particular cases. This was done by applying the general formalism developed in the chapter.
Walker, Uroyoan Ramon-emeterio, "On K-Conjugacy Classes of Maximal Tori in Semi-Simple Algebraic Groups." (2001). LSU Historical Dissertations and Theses. 370.