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Moufang Loops and Groups with Triality are Essentially the Same Thing
About this Title
J. I. Hall, Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 260, Number 1252
ISBNs: 978-1-4704-3622-3 (print); 978-1-4704-5321-3 (online)
DOI: https://doi.org/10.1090/memo/1252
Published electronically: July 16, 2019
Keywords: Moufang loop,
triality,
octonions
MSC: Primary 20N05
Table of Contents
Chapters
- Introduction
1. Basics
- 1. Category Theory
- 2. Quasigroups and Loops
- 3. Latin Square Designs
- 4. Groups with Triality
2. Equivalence
- 5. The Functor ${\mathbf {B}}$
- 6. Monics, Covers, and Isogeny in TriGrp
- 7. Universals and Adjoints
- 8. Moufang Loops and Groups with Triality are Essentially the Same Thing
- 9. Moufang Loops and Groups with Triality are Not Exactly the Same Thing
3. Related Topics
- 10. The Functors ${\mathbf {S}}$ and ${\mathbf {M}}$
- 11. The Functor ${\mathbf {G}}$
- 12. Multiplication Groups and Autotopisms
- 13. Doro’s Approach
- 14. Normal Structure
- 15. Some Related Categories and Objects
4. Classical Triality
- 16. An Introduction to Concrete Triality
- 17. Orthogonal Spaces and Groups
- 18. Study’s and Cartan’s Triality
- 19. Composition Algebras
- 20. Freudenthal’s Triality
- 21. The Loop of Units in an Octonion Algebra
Abstract
In 1925 Élie Cartan introduced the principal of triality specifically for the Lie groups of type $D_4$, and in 1935 Ruth Moufang initiated the study of Moufang loops. The observation of the title was made by Stephen Doro in 1978 who was in turn motivated by work of George Glauberman from 1968. Here we make the statement precise in a categorical context. In fact the most obvious categories of Moufang loops and groups with triality are not equivalent, hence the need for the word “essentially.”- M. Aschbacher, Finite group theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 2000. MR 1777008, DOI 10.1017/CBO9781139175319
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