The Conjugacy Classes of SL2 (IR)

Abstract

This paper is the result of my participation in the Undergraduate Program of the 1998 Park City Mathematics Institute Summer Session, as well as my individual work with Dr. John Fink for the remainder of that summer. The primary section of the paper, the SL2 (IR) problem, was proposed and outlined by Dr. Roger Howe, who lectured at the P.C.M.I. Summer Session. Superficially, the primary goal of this text is an exploration of the matrix group SL2(IR), the group of all 2 x 2 matrices with real entries and determinant 1, culminating with a geometric representation of its conjugacy classes. It is hoped, however, that the text might also demonstrate the usefulness of the basic concepts of Lie Theory in such investigations. For this reason, no previous knowledge of Lie Groups or Lie Algebras is presupposed of the reader. Rather, the text is designed to offer a brief introduction to these concepts within a setting comfortable to most undergraduate students. While it is assumed that the reader has a basic command of linear algebra, all other necessary results are developed or introduced within the text. The first section is a development of the connection between a Lie Group and its corresponding Lie Algebra. Rather than a rigorous development of this material, emphasis is placed on examples, and very few proofs are given. The results presented are only those which were manipulated in our explorations of SL(IR). The remainder of the text is devoted to these explorations. This portion begins by establishing the Lie Algebra of SL2 (IR), and the basic geometry of the algebra is determined. We then devote our attention to the conjugacy classes of the algebra, which are studied first algebraically, and then geometrically. The final sections examine the properties of the mapping from the algebra to the group SL2 (IR) and use this information to determine the geometry of the orbits of the group.