##
**The book of involutions. With a preface by J. Tits.**
*(English)*
Zbl 0955.16001

Colloquium Publications. American Mathematical Society (AMS). 44. Providence, RI: American Mathematical Society (AMS). xxi, 593 p. (1998).

The central theme of this book is the concept of algebras with involution. More precisely, let \(k\) be a commutative field, and let \(A\) be a finite-dimensional central simple algebra over \(k\). An involution \(\sigma\colon A\to A\) is an anti-automorphism of \(A\) such that \(\sigma^2\) is the identity. As pointed out by A. Weil (1960), algebras with involution are strongly related to classical groups. This connection is also at the heart of this book: it is not only the “book of involutions”, but also the “book of the classical groups”.

Beyond algebras with involution and classical groups, other related objects make their appearance in the later chapters. Indeed, the groups of type \(G_2\), \(F_4\), and also the trialitarian forms of the groups of type \(D_4\) are also related to certain algebras, such as Cayley algebras and Jordan algebras.

The “book of involutions” is a very welcome addition to the literature. The topics treated here have been the objects of very intensive recent research. The specialists felt the need of a reference book, and the beginners of a good introduction. Both these needs are fulfilled by the “book of involutions”.

The first Chapter is a very complete introduction to the notion of central simple algebra with involution, and the related ones of quadratic and Hermitian forms. In contrast to most texts on this topic, the ground field is not assumed to have characteristic different from two. The notion of quadratic pair, inspired by ideas of Tits, is used to give this uniform treatment.

The second Chapter is again a very complete description of existing results concerning the more classical type of invariants of algebras with involution, such as discriminants and Clifford algebras. Some of these are new, for instance the discriminant algebra of an involution of the second kind. From the cohomological point of view (made explicit in a later chapter of the book), these are related to \(H^1\) and \(H^2\) invariants. Again, the ideas of Tits are fundamental in this chapter. The invariants described here can be expressed as Tits algebras. This idea is explained in Chapter VI, together with several examples.

The connections with algebraic groups start to make their appearance – at first implicitly – in the third Chapter, where automorphisms of algebras with involution are discussed. The following two chapters are concerned with algebras of low degree. They contain many interesting examples, described in detail. The main idea of the first part of Chapter IV is to explain the so-called “exceptional isomorphisms” between classical groups from the point of view of algebras with involution. The second part is devoted to biquaternion algebras, and their many interesting properties. Chapter V focuses on algebras of degree 3. In particular, it contains a new presentation of results of Albert and a complete classification of unitary involutions of algebras of degree 3.

Chapter VI is a survey of the theory of linear algebraic groups over arbitrary fields. In particular, Weil’s correspondence between algebras with involution and the classical groups is described in detail. The last section of this chapter is concerned with the notion of Tits algebras, and its description in classical cases. With the tools introduced in this chapter, the reader will be able to understand some of the results of the earlier chapters from a different and more general point of view. The aim of Chapter VII is to recall some basic results concerning Galois cohomology. Several descent properties are described in full detail, and this is a very useful reference source. The last section of this chapter is a survey of Rost’s important results on \(H^3\)-invariants, unfortunately without proofs. As these results are not yet published elsewhere, this section will also be an important source for reference, and also an introduction to the subject.

The final three Chapters deal with the exceptional groups of type \(G_2\) and \(F_4\), and with the “exceptional classical groups”, the trialitarian forms of the groups of type \(D_4\). They give very useful surveys of results of Springer and Tits, and also a new notion, called trialitarian algebra. They give new ideas which should lead to better understanding of the phenomenon of triality.

The book is very well written. The first chapters are very detailed, give complete proofs and many examples. In the later chapters, the book is not always self-contained – this would have been impossible for a book of such a wide scope – but the authors are always careful to give the necessary references. The exercises and the historical comments are also very useful. In addition to being an excellent exposition of many basic results concerning algebras with involution and the classical groups, the book also contains many new ideas and new results, often due to the authors themselves. The topic is a very beautiful and vital one, object of intensive current research. This research is now made easier thanks to the impressive work of the four authors.

Beyond algebras with involution and classical groups, other related objects make their appearance in the later chapters. Indeed, the groups of type \(G_2\), \(F_4\), and also the trialitarian forms of the groups of type \(D_4\) are also related to certain algebras, such as Cayley algebras and Jordan algebras.

The “book of involutions” is a very welcome addition to the literature. The topics treated here have been the objects of very intensive recent research. The specialists felt the need of a reference book, and the beginners of a good introduction. Both these needs are fulfilled by the “book of involutions”.

The first Chapter is a very complete introduction to the notion of central simple algebra with involution, and the related ones of quadratic and Hermitian forms. In contrast to most texts on this topic, the ground field is not assumed to have characteristic different from two. The notion of quadratic pair, inspired by ideas of Tits, is used to give this uniform treatment.

The second Chapter is again a very complete description of existing results concerning the more classical type of invariants of algebras with involution, such as discriminants and Clifford algebras. Some of these are new, for instance the discriminant algebra of an involution of the second kind. From the cohomological point of view (made explicit in a later chapter of the book), these are related to \(H^1\) and \(H^2\) invariants. Again, the ideas of Tits are fundamental in this chapter. The invariants described here can be expressed as Tits algebras. This idea is explained in Chapter VI, together with several examples.

The connections with algebraic groups start to make their appearance – at first implicitly – in the third Chapter, where automorphisms of algebras with involution are discussed. The following two chapters are concerned with algebras of low degree. They contain many interesting examples, described in detail. The main idea of the first part of Chapter IV is to explain the so-called “exceptional isomorphisms” between classical groups from the point of view of algebras with involution. The second part is devoted to biquaternion algebras, and their many interesting properties. Chapter V focuses on algebras of degree 3. In particular, it contains a new presentation of results of Albert and a complete classification of unitary involutions of algebras of degree 3.

Chapter VI is a survey of the theory of linear algebraic groups over arbitrary fields. In particular, Weil’s correspondence between algebras with involution and the classical groups is described in detail. The last section of this chapter is concerned with the notion of Tits algebras, and its description in classical cases. With the tools introduced in this chapter, the reader will be able to understand some of the results of the earlier chapters from a different and more general point of view. The aim of Chapter VII is to recall some basic results concerning Galois cohomology. Several descent properties are described in full detail, and this is a very useful reference source. The last section of this chapter is a survey of Rost’s important results on \(H^3\)-invariants, unfortunately without proofs. As these results are not yet published elsewhere, this section will also be an important source for reference, and also an introduction to the subject.

The final three Chapters deal with the exceptional groups of type \(G_2\) and \(F_4\), and with the “exceptional classical groups”, the trialitarian forms of the groups of type \(D_4\). They give very useful surveys of results of Springer and Tits, and also a new notion, called trialitarian algebra. They give new ideas which should lead to better understanding of the phenomenon of triality.

The book is very well written. The first chapters are very detailed, give complete proofs and many examples. In the later chapters, the book is not always self-contained – this would have been impossible for a book of such a wide scope – but the authors are always careful to give the necessary references. The exercises and the historical comments are also very useful. In addition to being an excellent exposition of many basic results concerning algebras with involution and the classical groups, the book also contains many new ideas and new results, often due to the authors themselves. The topic is a very beautiful and vital one, object of intensive current research. This research is now made easier thanks to the impressive work of the four authors.

Reviewer: Eva Bayer-Fluckiger (BesanĂ§on)

### MSC:

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

12G05 | Galois cohomology |

11E39 | Bilinear and Hermitian forms |

11E57 | Classical groups |

11E72 | Galois cohomology of linear algebraic groups |

11E88 | Quadratic spaces; Clifford algebras |

16K20 | Finite-dimensional division rings |

17A75 | Composition algebras |

17C40 | Exceptional Jordan structures |

20G15 | Linear algebraic groups over arbitrary fields |