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Theory of Continuous Groups

Charles Loewner
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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This is a terrific little book! The question, however, is this: what is the audience? Theory of Continuous Groups is a rendering, by Harley Flanders and Murray Protter, of lecture notes from Charles Loewner’s 1955 course on the indicated material given during the latter’s visit to Berkeley (Loewner was at Stanford from 1950 till his death in 1968). As such, we have a set of very high quality presentations, aiming not so much at completeness as at providing a point of entry into the subject of continuous groups for gifted and motivated beginners, in a rather anachronistic style.

Is today’s rookie disposed to follow this sort of discussion? Is he willing and able to work through it all painstakingly, filling in gaps, following up leads, and so on? I am skeptical, I fear, because of the fact that contemporary American universities are inundated with so many texts on every nuance of every subject that a random would-be PhD finds himself stifled by too little formal context, so to speak.

It seems to me that in today’s pedagogical approach it would be unimaginable to present youngsters with a book on continuous groups coming in at fewer than, say, 300 pages. Loewner’s more holistic and exploratory approach, not aimed at all at completeness, seems out of place if not out of step. And, to boot, Theory of Continuous Groups sports an atypical historical element, given Loewner’s views on Lie’s original works (see below). All in all, it looks like the book would not resonate with today’s graduate students.

Well, more’s the pity. The core of Theory of Continuous Groups is a crystal-clear presentation of the theory of group representations — always of central importance — starting with finite groups and going all the way to compact groups (I’m always reminded of Holmes’ phrase to Watson in this connection: “The parallel is exact!”), with the rotation, orthogonal, and unitary examples properly featured.

Then it’s on to Lie theory, with a trio of fundamental theorems given very careful coverage. Interestingly, Loewner’s approach harkens back to nothing less than Sophus Lie’s original works, which, as Flanders and Protter point out, caused Loewner publically to express himself uncharacteristically caustically regarding Lie’s own presentation of this revolutionary material: evidently Loewner had to distill the indicated general results from Lie’s very rarified examples, as opposed to the main body of the expositions. Specifically, Loewner is apparently to be credited with genuine historical as well as mathematical scholarship in that he evolved “a coherent development of the subject by synthesizing [Lie’s] examples, many of which appeared only as footnotes.”

This said, there are in fact three “Fundamental Theorems” of Lie group theory in Loewner’s presentation, the first asserting, for example, that “two left-parallel vector fields of [a given Lie group] yield the same infinitesimal transformations of the realized transformation group” (p.82). This gives an indication of the book’s flavor.

It seems clear that the reader should have a decent background in algebra and analysis, at the advanced undergraduate of beginning graduate level, and he should possess sufficient mathematical maturity to be able (and willing) to get his hands very dirty very often, trying to come to grips with what is by current standards somewhat unusual mathematics: Loewner’s style is, as it were a generation or two out of date.

As I indicated above, while, ideally, we all reach the according stage of mathematical maturity at some point, it might all make for too much of an adaptation to anachronistic ways for today’s rookie. However, Loewner’s Theory of Continuous Groups will richly reward the committed and disciplined reader: in its own way it’s a rare gem.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Lecture I: Transformation Groups; Similarity
Lecture II: Representations of Groups; Combinations of Representations; Similarity and Reducibility
Lecture III: Representations of Cyclic Groups; Representations of Finite Abelian Groups; Representations of Finite Groups
Lecture IV: Representations of Finite Groups (cont.); Characters
Lecture V: Representations of Finite Groups (conc.); Introduction to Differentiable Manifolds; Tensor Calculus on a Manifold
Lecture VI: Quantities, Vectors, and Tensors; Generation of Quantities by Differentiation; Commutator of Two Contravariant Vector Fields; Hurwitz Integration on a Group Manifold
Lecture VII: Hurwitz Integration on a Group Manifold (cont.); Representation of Compact Groups; Existence of Representations
Lecture VIII: Representation of Compact Groups (cont.); Characters; Examples
Lecture IX: Lie Groups; Infinitesimal Transformations on a Manifold
Lecture X: Infinitesimal Transformations of a Group; Examples; Geometry on the Group Space
Lecture XI: Parallelism; First Fundamental Theorem of Lie Groups; Mayer-Lie Systems
Lecture XII: The Sufficiency Proof; First Fundamental Theorem; Converse; Second Fundamental Theorem; Converse
Lecture XIII: Converse of the Second Fundamental Theorem (cont.); Concept of Group Germ
Lecture XIV: Converse of the Third Fundamental Theorem; The Helmholtz-Lie Problem