Theory of Transformation Groups I: General Properties of Continuous Transformation Groups: A Contemporary Approach and Translation

Description

This modern translation of Sophus Lie's and Friedrich Engel's “Theorie der Transformationsgruppen Band I” will allow readers to discover the striking conceptual clarity and remarkably systematic organizational thought of the original German text. Volume I presents a comprehensive introduction to the theory and is mainly directed towards the generalization of ideas drawn from the study of examples. The major part of the present volume offers an extremely clear translation of the lucid original. The first four chapters provide not only a translation, but also a contemporary approach, which will help present day readers to familiarize themselves with the concepts at the heart of the subject.

The editor's main objective was to encourage a renewed interest in the detailed classification of Lie algebras in dimensions 1, 2 and 3, and to offer access to Sophus Lie's monumental Galois theory of continuous transformation groups, established at the end of the 19th Century. Lie groups are widespread in mathematics, playing a role in representation theory, algebraic geometry, Galois theory, the theory of partial differential equations, and also in physics, for example in general relativity.

This volume is of interest to researchers in Lie theory and exterior differential systems and also to historians of mathematics. The prerequisites are a basic knowledge of differential calculus, ordinary differential equations and differential geometry.

The following review is needlessly harsh. I found Merker's "mishmash" interesting; I especially liked Merker's frequent parenthetical quotes of the German original and the German index.

Reviewed by Fernando Q. Gouvêa, on 04/2/2015

Joël Merker’s main goal in this “modernized translation” is to give English-speaking mathematicians access to Sophus Lie’s famous Theorie der Transformationsgruppen , written with the collaboration of Friedrich Engel. It is clearly a labor of love: Merker tells us that he learned German specifically to be able to read Lie, and that he came to the conclusion that “ Lie’s mathematical thought is universal and transhistorical ” (italics in the original). The result is very useful, though it does have some quirks.

In the preface, Merker says that

For several reasons, it was essentially impossible to directly translate the first few chapters in which Lie’s intention was to set up the beginnings of the theory in the highest possible generality, especially in order to eliminate the axiom of inverse, an aspect never dealt with in modern treatises. As a result, I decided in the first four chapters to reorganize the material and reprove the relevant statements, nevertheless retaining all the mathematical content.

Of course, it is clearly possible to translate the first few chapters! What Merker means, I guess, is that it is impossible to make sense of those chapters as they stand, and so he has provided a modern account with proofs that are acceptable to us. As a result, these chapters are a mishmash of Merker and Lie-Engel. Some sections are shaded in gray, but even there it is not clear whether we are reading Lie himself or Merker’s modernization. (I may have missed it, but it seems to me that the meaning of the gray shading is never specified.) I would have preferred to see a straight translation next to modern commentary, perhaps on facing pages. Historians interested in what Lie actually said will have to compare the text with the original.

Luckily,

But starting with Chap. 5, Engel and Lie’s exposition is so smooth, so rigorous, so understandable, so systematic, so astonishingly well organized — so beautiful for thought — that a pure translation is essential.

Well, almost. There are footnotes, which may or may not be Merker’s. Most chapters end with a horizontal rule, but in some chapters that rule is followed by text which seems to be commentary. And the gray shading reappears at times, mysteriously. There are also some moments when a footnote marker appears, but there is no footnote; instead, a translator’s note appears in parentheses. Like this, from page 261:

In addition, since the mentioned equations are certainly compatible with each other,1 (Translator’s note: — and since, furthermore, the lemma on p. 78 insures that, with a suitable choice of generic fixed points x′1,…,x′n,x′′1,…,x′′n,⋯,x(q)1,…x(q)n

, the rank of the considered matrix of ξ’s is maximal equal to q

— ) we obtain…

To summarize: Merker has provided us with a valuable translation of a crucially important text, but Springer could have done a better job of production, and both Merker and Springer should have been more careful to distinguish the commentary from the text.