Lie group theory is a topic every mathematician should be familiar with. Its range of applications includes physics, differential equations, number theory, and symplectic manifolds.
A Lie group is a smooth manifold, real or complex, endowed with a group structure such that the multiplication and the inversion are smooth maps. The easiest examples are the real line, the circle S1 , the multiplicative group of the nonzero real numbers. An important source of examples of Lie groups is the family of classical groups, that is, the groups of transformations of a finite-dimensional real or complex vector space, leaving invariant some fixed quadratic form. For example: the unitary, symplectic or orthogonal groups are classical Lie groups.
As the title claims, this book focuses on a particular class of Lie groups: the linear ones. These are the Lie groups obtained as subgroups of some GL(n). They do not exhaust all the Lie groups: the so-called exceptional groups are not linear groups.
The relevance of Lie groups to physics is clear: a linear transformation of a real or complex finite-dimensional space can be thought of as a vector field and linear Lie groups can be thought of as groups of symmetries of dynamical systems.
The author’s purpose is to offer an introduction to many topics in Lie group theory, assuming as prerequisites only a basic course in calculus and linear algebra; in particular, no knowledge of differential geometry is required in order to read the book. This quest for simplicity is the main reason to restrict the exposition to linear groups, because in this case the differential structure can be given explicitly through the exponential map on matrices. More generally, the exponential map plays a central role throughout the whole book (for example, the Lie correspondence for connected groups is proved as a consequence of the
A nice feature of this book is a variety of problems proposed at the end of each section; this makes it especially suitable as a textbook for a first course on Lie groups, addressed to an audience of mathematicians or physicists.
Groups are often better understood through their representations and Lie groups are not exceptions to this rule; of course, the representations to be considered are smooth representations. Chapter 6 introduces the reader to some all-important results on representations of Lie groups: the Peter-Weyl theorem, Weyl’s formula for characters of unitary groups, and many others are the object of the exercises.
There is a huge amount of books on Lie groups, taking different approaches, the most famous being probably Knapp’s books (Lie Groups: Beyond An Introduction, Representation Theory of Semisimple Lie Groups), of exceptional quality but more difficult to read. Other elementary introductions exist, but to my knowledge, none is as complete as this one, covering arguments as the fundamental groups, root systems, integration theory, and with a number of well-chosen examples. As the author suggests, one could well combine a course based on this text, with a course on Lie algebras, based for example on Humphreys’ book, Introduction to Lie Algebras and Representation Theory.
The book provides all the tools to proceed further in the subject, so the interested reader will not find any major difficulty moving on to understand Dynkin diagrams and exceptional groups. In fact, exceptional groups are also important in physical applications, especially in the theory of superstrings (for example, the bosonic symmetry for the five-dimensional supergravity, corresponds to the exceptional Lie group called E6).
Besides the applications to physics, Lie groups, and especially their representations, are among the most useful and beautiful theories in mathematics; there is also a particularly interesting connection with number theory, through the theory of modular forms or, more precisely, automorphic forms. The number-theoretic importance of Lie groups is encoded in the Langlands correspondence (see http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/intro.html ).
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at firstname.lastname@example.org.
1. The exponential map
2. Lie theory
3. The classical groups
4. Manifolds, homogeneous spaces, Lie groups
Appendix: Analytic Functions and Inverse Function Theorem