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Exceptional Lie algebras. (English) Zbl 0215.38701

Lecture Notes in Pure and Applied Mathematics. 1. New York: Marcel Dekker, Inc. v, 125 p. (1971).
“Exceptional Lie algebras” was the title of a set of mimeographed notes by the author which have since 1958 been part of the host of secret papers circulating in the mathematical world, known only to a restricted number of specialists in the field who had been so lucky as to obtain a copy. These notes have now been made available to the general mathematical public due to the present fashion of Publishers who have started distributing offset reproductions of typewritten originals.
The present edition has more than twice the size of the original notes; the part which has been added contains recent distributions to the field by the author and some of his former students (H. P. Allen, J. C. Ferrar) and by J. Tits, M. Koecher and others. Thus the booklet provides a complete description of a set of models for the exceptional Lie algebras \(G_2\), \(F_4\), \(E_8\), \(E_7\) and \(E_6\) over algebraically closed fields of characteristic 0 and over the reals, and of exceptional forms of \(D_4\). These models are all obtained in one way or another by the use of Cayley algebras and exceptional Jordan algebras.
The first three sections provide the basic material on these algebras, their derivation algebras and some subalgebras thereof, thus introducing realizations of Lie algebras of type \(D_4\), \(F_4\) and \(E_6\). The following two sections are devoted to a study of Lie algebras of type \(D_4\). §6 displays the roots of \(F_4\) and \(E_7\), whereas §7 deals with the Lie algebra \(E_6\). These seven sections form a reproduction of the original 1958 notes with a few minor changes.
§8 gives applications of Galois cohomology to forms of \(E_6\) such as they have appeared in Ferrar’s thesis. The next two sections are devoted to Tits’ first and second construction, respectively, which yield descriptions for Lie algebras of type \(E_7\) and \(E_6\) (among many others ones). The form in which \(E_7\) is presented here is closely related to Koecher’s construction. §11 gives methods to compute the Killing form of the Lie algebras considered here. Finally, in §12 a complete list is given of the real forms of the exceptional simple Lie algebras.
The book shows the signs of publication in a provisional form. For instance, it is not always easy to find a particular result in it, the absence of an index being only partly compensated by the table of contents. Nevertheless, for anyone interested in this field this is a very valuable publication which shows the author’s mastery of the subject as well as his clear style.

MSC:

17B25 Exceptional (super)algebras
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras