##
**Octonions, Jordan algebras and exceptional groups.
Revised English version of the original German notes.**
*(English)*
Zbl 1087.17001

Springer Monographs in Mathematics. Berlin: Springer (ISBN 3-540-66337-1/hbk). viii, 208 p. (2000).

This book is an updated and revised English version of the German notes on octaves, Jordan algebras and exceptional groups which appeared as mimeographed lecture notes of GĂ¶ttingen University in 1963. It is still an excellent reference on the subject which is now available again.

The first chapter contains the basic theory of composition algebras without restrictions on the characteristic of the underlying field \(k\). Also a classification of composition algebras over some special fields is given.

In the second chapter the group \(\text{Aut}(C)\) of (linear) automorphisms of an octonion algebra \(C\) over a field of arbitrary characteristic is studied. \(\text{Aut}(C)\) is a subgroup of the orthogonal group \(O(N)\) of the norm \(N\) of \(C\). Let \(K\) be an algebraic closure of \(k\) and let \(C_K = K\otimes_k C\). Then the automorphism group \({\mathbf G} = \text{Aut}(C_K)\) is a linear algebraic group. \(G\) is also a closed subgroup of the algebraic group \(\mathbf O(N)\). The group \(\mathbf G\) is identified as a connected, simple algebraic group of type \(G_2\). Moreover, it is shown that \(\mathbf G\) is defined over \(k\). Thus the automorphism group \(G\) is the group \({\mathbf G}(k)\) of \(k\)-rational points of the algebraic group \(\mathbf G\).

Chapter 3 deals with algebraic triality in the group of similarities and in the orthogonal group \(O(N)\) of the norm \(N\) of an octonion algebra \(C\). Moreover, the related triality in the Lie algebras of these groups, the so-called local triality, is considered.

Chapter 4 focuses on twisted composition algebras. First twisted composition algebras are considered in the case, that the separable cubic extension \(t\) of the underlying field \(k\) is normal, hence a cubic cyclic extension. In this case the characteristic of \(k\) can be arbitrary. These algebras are called “normal twisted composition algebras” and were introduced by T. A. Springer. The concept of twisted composition algebras has been generalized by F. D. Veldkamp to the case that the separable cubic extension of \(k\) is not normal. Here the characteristics 2 and 3 have to be excluded. In both cases one has a vector space with an additional operation which is binary in the normal case and unary in the nonnormal case.

Also the automorphism groups of twisted composition algebras are studied and some explicit constructions of twisted composition algebras are given.

In chapter 5 a class of Jordan algebras is discussed which includes the exceptional central simple Jordan algebras or Albert algebras. Albert algebras are connected with exceptional simple groups of type \(E_6\) and \(F_4\). Here only the so-called \(J\)-algebras are considered and described by simple axioms. Thereby characteristic 3 is excluded for technical reasons, but characteristic \(\neq 2\) is essential for the approach given here. Thus a description of all reduced \(J\)-algebras is obtained.

Chapter 6 develops another description of \(J\)-algebras including the nonreduced ones. For this purpose \(J\)-algebras and twisted composition algebras are linked together. Thus a \(J\)-algebra is reduced if and only if certain twisted composition algebras are reduced. In particular, every \(J\)- algebra over an algebraic number field is reduced.

Chapter seven deals with some exceptional groups which are associated with Albert algebras. First the automorphism group of an Albert algebra is determined. It is shown that this is an exceptional simple group oftype \(F_4\). Then the group of transformations leaving the cubic form det invariant is studied and is identified as a group of type \(E_6\). The underlying fields are supposed to be of characteristic \(\neq 2,3\).

The concluding chapter eight gives a sketchy survey on more recent results in the theory of octonion and Albert algebras. First a rudimentary exposition of some notions from Galois cohomology is given. Then some cohomological invariants of composition algebras, especially of octonion algebras, of twisted octonion algebras and of Albert algebras are considered. Finally the relation between the decomposition of an Albert division algebra \(A\) into subspaces and the Freudenthal-Tits-construction (or first Tits construction) is indicated.

The first chapter contains the basic theory of composition algebras without restrictions on the characteristic of the underlying field \(k\). Also a classification of composition algebras over some special fields is given.

In the second chapter the group \(\text{Aut}(C)\) of (linear) automorphisms of an octonion algebra \(C\) over a field of arbitrary characteristic is studied. \(\text{Aut}(C)\) is a subgroup of the orthogonal group \(O(N)\) of the norm \(N\) of \(C\). Let \(K\) be an algebraic closure of \(k\) and let \(C_K = K\otimes_k C\). Then the automorphism group \({\mathbf G} = \text{Aut}(C_K)\) is a linear algebraic group. \(G\) is also a closed subgroup of the algebraic group \(\mathbf O(N)\). The group \(\mathbf G\) is identified as a connected, simple algebraic group of type \(G_2\). Moreover, it is shown that \(\mathbf G\) is defined over \(k\). Thus the automorphism group \(G\) is the group \({\mathbf G}(k)\) of \(k\)-rational points of the algebraic group \(\mathbf G\).

Chapter 3 deals with algebraic triality in the group of similarities and in the orthogonal group \(O(N)\) of the norm \(N\) of an octonion algebra \(C\). Moreover, the related triality in the Lie algebras of these groups, the so-called local triality, is considered.

Chapter 4 focuses on twisted composition algebras. First twisted composition algebras are considered in the case, that the separable cubic extension \(t\) of the underlying field \(k\) is normal, hence a cubic cyclic extension. In this case the characteristic of \(k\) can be arbitrary. These algebras are called “normal twisted composition algebras” and were introduced by T. A. Springer. The concept of twisted composition algebras has been generalized by F. D. Veldkamp to the case that the separable cubic extension of \(k\) is not normal. Here the characteristics 2 and 3 have to be excluded. In both cases one has a vector space with an additional operation which is binary in the normal case and unary in the nonnormal case.

Also the automorphism groups of twisted composition algebras are studied and some explicit constructions of twisted composition algebras are given.

In chapter 5 a class of Jordan algebras is discussed which includes the exceptional central simple Jordan algebras or Albert algebras. Albert algebras are connected with exceptional simple groups of type \(E_6\) and \(F_4\). Here only the so-called \(J\)-algebras are considered and described by simple axioms. Thereby characteristic 3 is excluded for technical reasons, but characteristic \(\neq 2\) is essential for the approach given here. Thus a description of all reduced \(J\)-algebras is obtained.

Chapter 6 develops another description of \(J\)-algebras including the nonreduced ones. For this purpose \(J\)-algebras and twisted composition algebras are linked together. Thus a \(J\)-algebra is reduced if and only if certain twisted composition algebras are reduced. In particular, every \(J\)- algebra over an algebraic number field is reduced.

Chapter seven deals with some exceptional groups which are associated with Albert algebras. First the automorphism group of an Albert algebra is determined. It is shown that this is an exceptional simple group oftype \(F_4\). Then the group of transformations leaving the cubic form det invariant is studied and is identified as a group of type \(E_6\). The underlying fields are supposed to be of characteristic \(\neq 2,3\).

The concluding chapter eight gives a sketchy survey on more recent results in the theory of octonion and Albert algebras. First a rudimentary exposition of some notions from Galois cohomology is given. Then some cohomological invariants of composition algebras, especially of octonion algebras, of twisted octonion algebras and of Albert algebras are considered. Finally the relation between the decomposition of an Albert division algebra \(A\) into subspaces and the Freudenthal-Tits-construction (or first Tits construction) is indicated.

Reviewer: Huberta Lausch (Grafing)

### MathOverflow Questions:

An isomorphic classification of non-associative division octonion algebrasThe octonions on a bad day

### MSC:

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

17A75 | Composition algebras |

17C10 | Structure theory for Jordan algebras |

17C30 | Associated groups, automorphisms of Jordan algebras |

20G15 | Linear algebraic groups over arbitrary fields |