This work provides a first taste of the theory of Lie groups accessible to advanced mathematics undergraduates and beginning graduate students, providing an appetiser for a more substantial further course. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory. We concentrate on matrix groups, i.e., closed subgroups of real and complex general linear groups. One of the results proved is that every matrix group is in fact a Lie group, the proof following that in the expository paper of Howe [12]. Indeed, the latter, together with the book of Curtis [7], influenced our choice of goals for the present book and the course which it evolved from. As pointed out by Howe, Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra, and exposure to them can inform or motivate the study of the latter; we frequently describe such topics in enough detail to provide the necessary background for the benefit of readers unfamiliar with them.
Outline of the Chapters
Each chapter contains exercises designed to consolidate and deepen readers' understanding of the material covered. We also use these to explore related topics that may not be familiar to all readers but which should be in the toolkit of every well-educated mathematics graduate. Here is a brief synopsis of the chapters.
Chapter 1: The general linear groups GL„(k) for к = R (the real numbers) and к = С (the complex numbers) are introduced and studied both as groups and as topological spaces. Matrix groups are defined and a number of standard examples discussed, including special linear groups SLn(k), orthogonal groups O(n) and special orthogonal groups SO(n), unitary groups U(n) and special unitary groups SU(rc), as well as more exotic examples such as Lorentz groups
