Moufang Loops and Groups with Triality Are Essentially the Same Thing.

Cover -- Title page -- Introduction -- Part 1 . Basics -- Chapter 1. Category Theory -- 1.1. Basics -- 1.2. Category equivalence -- 1.3. Terminal objects and kernel morphisms -- 1.4. Pointed categories -- 1.5. Rank 1 objects -- 1.6. Simplicity -- Chapter 2. Quasigroups and Loops -- 2.1. Basics -- 2.2. Autotopisms and anti-autotopisms -- 2.3. Loop homomorphisms and the pointed category \aLoop -- 2.4. Moufang loops and other loop varieties -- 2.5. Examples -- Chapter 3. Latin Square Designs -- 3.1. Basics -- 3.2. Central Latin square designs -- 3.3. The correspondence between \Mouf and \CLSD -- 3.4. Cayley tables of groups -- Chapter 4. Groups with Triality -- 4.1. Basics -- 4.2. Examples -- 4.3. Normal subgroups in the base group and wreath products -- Part 2 . Equivalence -- Chapter 5. The Functor \functor{ } -- 5.1. A presentation -- 5.2. The functor \dthominvname -- Chapter 6. Monics, Covers, and Isogeny in \catTriGrp -- 6.1. A fibered product -- 6.2. Monics in \Tri -- 6.3. Covers and isogeny -- Chapter 7. Universals and Adjoints -- 7.1. Universal and adjoint groups -- 7.2. Universal and adjoint categories -- Chapter 8. Moufang Loops and Groups with Triality are Essentially the Same Thing -- 8.1. A category equivalence -- 8.2. Monics -- Chapter 9. Moufang Loops and Groups with Triality are Not Exactly the Same Thing -- 9.1. \Mouf and \Tri are not equivalent -- 9.2. \aMouf and \aTri are not equivalent -- 9.3. \aMouf and \aATri are not equivalent -- Part 3 . Related Topics -- Chapter 10. The Functors \functor{ } and \functor{ } -- 10.1. \thominvname and \anchor{\thominvname} -- 10.2. \lthomname and \anchor{\lthomname} -- Chapter 11. The Functor \functor{ } -- 11.1. \gthomname and \anchor{\gthomname} -- 11.2. Properties of universal groups -- 11.3. Another presentation -- Chapter 12. Multiplication Groups and Autotopisms.

12.1. Multiplication and inner mapping groups -- 12.2. Autotopisms -- 12.3. Moufang multiplication groups, nuclei, and special autotopisms -- Chapter 13. Doro's Approach -- 13.1. Doro's categories -- 13.2. A presentation of the base group -- 13.3. Equivalent presentations -- 13.4. Moufang loops -- Chapter 14. Normal Structure -- 14.1. Simplicity -- 14.2. Short exact sequences -- 14.3. Solvable Moufang loops -- Chapter 15. Some Related Categories and Objects -- 15.1. 3-nets -- 15.2. Categories of conjugates -- 15.3. Groups enveloping triality -- 15.4. Tits' symmetric \calT-geometries -- 15.5. Latin chamber systems covered by buildings -- Part 4 . Classical Triality -- Chapter 16. An Introduction to Concrete Triality -- 16.1. Study's triality -- 16.2. Cartan's triality -- 16.3. Composition algebras and the octonions -- 16.4. Freudenthal's triality -- 16.5. Moufang loops from octonion algebras -- Chapter 17. Orthogonal Spaces and Groups -- 17.1. Orthogonal geometry -- 17.2. Hyperbolic orthogonal spaces -- 17.3. Oriflamme geometries -- 17.4. Orthogonal groups -- 17.5. Chevalley groups \oD_{ }( ) -- 17.6. Orthogonal groups in dimension 8 -- Chapter 18. Study's and Cartan's Triality -- 18.1. Triality geometries and Study's triality -- 18.2. Cartan's triality -- Chapter 19. Composition Algebras -- 19.1. Composition algebras -- 19.2. General structure -- 19.3. Hurwitz' Theorem in a restricted version -- 19.4. Doubling and Hurwitz' Theorem in its general version -- 19.5. Commuting and associating -- 19.6. Algebraic triality for the split octonions -- Chapter 20. Freudenthal's Triality -- 20.1. Some calculations -- 20.2. Freudenthal's triality -- 20.3. The spin kernel and spin group -- Chapter 21. The Loop of Units in an Octonion Algebra -- 21.1. Loop sections of octonion algebras -- 21.2. Octonion multiplication and triality groups.

21.3. The split octonions -- 21.4. Simple Moufang loops -- Bibliography -- Index -- Back Cover.

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 in 1978 was made by Stephen Doro, who was in turn motivated by the work of George Glauberman from 1968. Here the author makes 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.".

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.