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Author Hall, J. I., author

Title Moufang loops and groups with triality are essentially the same thing / J.I. Hall
Published Providence, RI : American Mathematical Society, 2019


Description 1 online resource (xiv, 186 pages)
Series Memoirs of the American Mathematical Society, 1947-6221 ; volume 260, number 1252
Memoirs of the American Mathematical Society ; no. 1252
Contents 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 Triality4.1. 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 Thing9.1. \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 groups12.2. 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 buildingsPart 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
Summary In 1925 √Člie 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.""
Bibliography Includes bibliographical references and index
Notes Online resource; title from PDF title page (viewed August 27, 2019)
Subject Moufang loops.
Cayley numbers (Algebra)
Cayley numbers (Algebra)
Moufang loops.
Form Electronic book
ISBN 1470453215