J. Frank Adams was internationally known and respected as one of the great algebraic topologists. Adams had long been fascinated with exceptional Lie groups, about which he published several papers, and he gave a series of lectures on the topic. The author's detailed lecture notes have enabled volume editors Zafer Mahmud and Mamoru Mimura to preserve the substance and character of Adams's work.
Because Lie groups form a staple of most mathematics graduate students' diets, this work on exceptional Lie groups should appeal to many of them, as well as to researchers of algebraic geometry and topology.
J. Frank Adams was Lowndean professor of astronomy and geometry at the University of Cambridge. The University of Chicago Press published his Lectures on Lie Groups and has reprinted his Stable Homotopy and Generalized Homology .
Chicago Lectures in Mathematics Series
Contents/Summary......Page 4
Summary of Constructions......Page 9
Foreword......Page 10
Acknowledgments......Page 11
Introduction......Page 12
Chapter 1. Definitions, examples and matrix groups......Page 14
Infinitesimal methods......Page 15
Representation theory of compact groups......Page 16
Weights and characters......Page 17
Sketch of classification of compact Lie groups......Page 18
Chapter 2. Clifford algebras......Page 26
Structure maps on Clifford algebras......Page 28
Chapter 3. The Spin groups......Page 30
Chapter 4. Clifford modules and representations......Page 34
The theorem of Weyl on R{G)......Page 41
Construction of $G_2$......Page 44
$Spin(8)$ and triality......Page 46
Chapter 6. The exceptional groups: construction of $E_8$......Page 50
Construction of a Lie algebra of type $E_8$......Page 51
Standard operating procedure......Page 53
The Killing form......Page 55
Chapter 7. Construction of a Lie group of type $E_8$......Page 58
Real forms of $E_8$......Page 59
Chapter 8. The construction of Lie groups of type $F_4$, $E_6$, $E_7$......Page 62
Identification of the subgroups $H$......Page 63
Identification of $L(G)$ and $L(G)/L(H)$......Page 64
Real forms of $E_8$, continued......Page 66
$F_4$......Page 68
$E_7 \\times SU/(2)/Z_2$......Page 69
$E_6 \\times SU/(3)$......Page 70
Chapter 10. The Weyl group of $E_8$......Page 72
Chapter 11. Representations of $E_6$, $E_7$......Page 82
Chapter 12. Direct construction of $E_7$......Page 86
Construction of $L(E_7)$ and its 56-dimensional representation......Page 87
$E_7$ as a group of maps of $W$ (Cartan's construction)......Page 93
Real forms of $E_7$......Page 95
Construction of $E_6$ and its 27-dimensional representation......Page 98
$E_6$ as a group of maps......Page 104
Chapter 14. Direct treatment of $F_4$, I......Page 106
Structure maps on $U$......Page 109
The algebra structure on $U$......Page 113
Chapter 15. The Cayley numbers......Page 118
Connection between the Cayley numbers and Lie groups......Page 124
Definition and properties of the exceptional Jordan algebra $J$......Page 126
The Cayley projective plane......Page 131
Appendix. Jordan algebras......Page 132
References......Page 134
