**Memoirs of the American Mathematical Society**

2005;
99 pp;
Softcover

MSC: Primary 51; 16; 20;

Print ISBN: 978-0-8218-3608-8

Product Code: MEMO/173/818

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-0419-2
Product Code: MEMO/173/818.E**

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

# An Algebraic Structure for Moufang Quadrangles

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*Tom De Medts*

Very recently, the classification of Moufang polygons has been completed by Tits and Weiss. Moufang \(n\)-gons exist for \(n \in \{ 3, 4, 6, 8 \}\) only. For \(n \in \{ 3, 6, 8 \}\), the proof is nicely divided into two parts: first, it is shown that a Moufang \(n\)-gon can be parametrized by a certain interesting algebraic structure, and secondly, these algebraic structures are classified. The classification of Moufang quadrangles \((n=4)\) is not organized in this way due to the absence of a suitable algebraic structure. The goal of this article is to present such a uniform algebraic structure for Moufang quadrangles, and to classify these structures without referring back to the original Moufang quadrangles from which they arise, thereby also providing a new proof for the classification of Moufang quadrangles, which does consist of the division into these two parts. We hope that these algebraic structures will prove to be interesting in their own right.

#### Readership

Graduate students and research mathematicians interested in algebra and algebraic geometry.

#### Table of Contents

# Table of Contents

## An Algebraic Structure for Moufang Quadrangles

- Contents v6 free
- Chapter 1. Introduction 18 free
- Acknowledgment 29 free

- Chapter 2. Definition 310
- Chapter 3. Some Identities 714
- Chapter 4. From Quadrangular Systems To Moufang Quadrangles 1522
- Chapter 5. From Moufang Quadrangles To Quadrangular Systems 2330
- Chapter 6. Some Remarks 3542
- Chapter 7. Examples 3744
- 7.1. Quadrangular Systems of Quadratic Form Type 3744
- 7.2. Quadrangular Systems of Involutory Type 3744
- 7.3. Quadrangular Systems of Indifferent Type 3845
- 7.4. Quadrangular Systems of Pseudo- quadratic Form Type 3946
- 7.5. Quadrangular Systems of Type E[sub(6)], E[sub(7)] and E[sub(8)] 4047
- 7.6. Quadrangular Systems of Type F[sub(4)] 4350

- Chapter 8. The Classification 4754
- 8.1. Quadrangular Systems of Involutory Type 4956
- 8.2. Quadrangular Systems of Quadratic Form Type 6067
- 8.3. Quadrangular Systems of Indifferent Type 6370
- 8.4. Quadrangular Systems of Pseudo-quadratic Form Type, I 6572
- 8.5. Quadrangular Systems of Type F[sub(4)] 7077
- 8.6. Quadrangular Systems of Pseudo-quadratic Form Type, II 8188
- 8.7. Quadrangular Systems of Type E[sub(6)], E[sub(7)] and E[sub(8)] 93100

- Appendix A. Abelian Quadrangular Systems 95102
- Bibliography 99106