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Softcover ISBN:  9781470436223 
Product Code:  MEMO/260/1252 
List Price:  $81.00 
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AMS Member Price:  $48.60 
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Product Code:  MEMO/260/1252.E 
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AMS Member Price:  $48.60 
Softcover ISBN:  9781470436223 
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Product Code:  MEMO/260/1252.B 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 260; 2019; 186 ppMSC: Primary 20
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.”

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


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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.”

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