Lie Groups: An Introduction Through Linear Groups (Oxford Graduate Texts in Mathematics, Number 5)

Rating:
Summary: Rossmann Achieves the Impossible!
Review: Absolutely Fabulous (apologies to Dawn French!). It is truly a privilege to read this book! Rossmann has achieved the impossible and given me a firm grip on Lie theory. This he does within the framework of matrix groups (familiar examples of manifolds) - he mounts his trusted steed, Exponential Map, and travels very light through the wonderful lands he explores - the Inverse Function Theorem, the Baker Campbell Hausdorf formula, and basic group / linear space properties are really the only four spells he takes with him in his kit bag. His steed bears him to the Lie Correspondence (Lie's third theorem) with truly dazzling speed and simplicity. Many if not most of the proofs and results throughout the book fall back on very first principles, so you won't need a Cetacean-sized brain to keep abreast of his discussion. Nonetheless, there is no sacrifice of rigour. The book clearly shows that Rossmann is highly experienced in the teaching of his subject matter - the proofs all have a highly polished and pared feel to them; they are clearly refined by many years of questionings by floundering students and newcomers such as myself because very few items are not apparent at a first or second reading.

The latter part of the book goes onto the conventional (manifold-oriented) definition of a Lie group. By this stage, the reader is superbly equipped, and thus the first four chapters of the book would, I believe, serve as an excellent inroad into differential geometry (which Rossmann also has a book about). The book is repleat with splendid and interesting examples and the style down-to-earth: there is no haughty trivialisation of concepts that Rossmann readily acknowledges are awkward at first reading.

Incidentally, my need for Lie theory is through the solution of the Master Equation for coupled optical waveguides. The set of transfer matrices generally forms a Lie group and the Lie algebra is the set of coupling matrices, which describe the cross-sectional geometry and dielectric properties of the coupled system. One-parameter subgroups are then sets of translationally invariant waveguide systems, and the transfer matrices then the exponentials of the invariant coupling matrices. Rossmann's approach through the Exponential Map is thus particularly well suited for the acquisition of the Lie machinery needed to study such systems.

A final note: make sure you get hold of the 2003 reprinting, which corrects some unfortunate typos that would otherwise slightly marr an excellent learning adventure.