The classification of semisimple algebraic groups. Collected works. Vol. 3. With the collaboration of P. Cartier, A. Grothendieck and M. Lazard. Text revised by P. Cartier.
(Classification des groupes algébriques semi-simples.)

*(French)*Zbl 1099.01026
Berlin: Springer (ISBN 3-540-23031-9/hbk). xiii, 276 p. (2005).

This third volume of the collected works of Claude Chevalley is different. It is a reissue of the Séminaire Chevalley of 1956/57 and 1957/58. So it is only partly written by Chevalley himself. The other authors are P. Cartier, A. Grothendieck and M. Lazard.

Recall that in those seasons the seminar was devoted to developing the theory of algebraic groups in arbitrary characteristic, up to the point of proving uniqueness in the classification of semisimple algebraic groups over an algebraically closed field. Existence is not covered for all Dynkin types. It is relegated to T. Chevalley’s memoir [“Sur certains groupes simples”. Tohoku Math. J., II. Ser. 7, 14–66 (1955; Zbl 0066.01503)].

The text of the seminar has now been typeset in T

Of course it is not a textbook. It is a seminar. No great effort has been made to minimize the prerequisites, or to limit the discussion to what will be needed later. At its high level the text is rather self-contained. A big part of the theory is due to Chevalley himself. The starting point is work of Kolchin and Borel on algebraic groups, also explained here. But first we have to learn some algebraic geometry. Some things are a bit dated, like the convention that varieties are irreducible. This convention is a nuisance, as some important closed subgroups of algebraic groups are not connected. Also, the schemes are still reduced in this work. But what strikes much more than such quaintness is how relevant the original treatment still is. The Borel subgroups, the Weyl group, the line bundles on the flag variety, a construction of homogeneous spaces, Jordan decomposition, singular tori, isogenies …, they are all here, treated in full detail.

Recall that in those seasons the seminar was devoted to developing the theory of algebraic groups in arbitrary characteristic, up to the point of proving uniqueness in the classification of semisimple algebraic groups over an algebraically closed field. Existence is not covered for all Dynkin types. It is relegated to T. Chevalley’s memoir [“Sur certains groupes simples”. Tohoku Math. J., II. Ser. 7, 14–66 (1955; Zbl 0066.01503)].

The text of the seminar has now been typeset in T

_{E}X. Then it was edited by P. Cartier. He has made minor corrections and added small things like footnotes, captions, a Postface. But basically this is the original seminar. After 50 years this is still a text to recommend to anyone who reads French and wants to learn more about algebraic groups.Of course it is not a textbook. It is a seminar. No great effort has been made to minimize the prerequisites, or to limit the discussion to what will be needed later. At its high level the text is rather self-contained. A big part of the theory is due to Chevalley himself. The starting point is work of Kolchin and Borel on algebraic groups, also explained here. But first we have to learn some algebraic geometry. Some things are a bit dated, like the convention that varieties are irreducible. This convention is a nuisance, as some important closed subgroups of algebraic groups are not connected. Also, the schemes are still reduced in this work. But what strikes much more than such quaintness is how relevant the original treatment still is. The Borel subgroups, the Weyl group, the line bundles on the flag variety, a construction of homogeneous spaces, Jordan decomposition, singular tori, isogenies …, they are all here, treated in full detail.

Reviewer: Wilberd van der Kallen (Utrecht)