Jordan Algebras and Algebraic Groups

1. Front Matter

   Pages I-VII

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2. Preliminaries

   • Tonny A. Springer

   Pages 1-8

3. J-structures

   • Tonny A. Springer

   Pages 9-22

4. Examples

   • Tonny A. Springer

   Pages 23-38

5. The Quadratic Map of a J-structure

   • Tonny A. Springer

   Pages 39-47

6. The Lie Algebras Associated with a J-structure

   • Tonny A. Springer

   Pages 48-53

7. J-structures of Low Degree

   • Tonny A. Springer

   Pages 54-65

8. Relation with Jordan Algebras (Characteristic ≠2)

   • Tonny A. Springer

   Pages 66-71

9. Relation with Quadratic Jordan Algebras

   • Tonny A. Springer

   Pages 72-78

10. The Minimum Polynomial of an Element

    • Tonny A. Springer

    Pages 79-82

11. Ideals, the Radical

    • Tonny A. Springer

    Pages 83-89

12. Peirce Decomposition Defined by an Idempotent Element

    • Tonny A. Springer

    Pages 90-105

13. Classification of Certain Algebraic Groups

    • Tonny A. Springer

    Pages 106-121

14. Strongly Simple J-structures

    • Tonny A. Springer

    Pages 122-127

15. Simple J-structures

    • Tonny A. Springer

    Pages 128-135

16. The Structure Group of a Simple J-structure and the Related Lie Algebras

    • Tonny A. Springer

    Pages 136-157

17. Rationality Questions

    • Tonny A. Springer

    Pages 158-165

18. Back Matter

    Pages 167-173

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From the reviews: "This book presents an important and novel approach to Jordan algebras. Jordan algebras have come to play a role in many areas of mathematics, including Lie algebras and the geometry of Chevalley groups. Springer's work will be of service to research workers familiar with linear algebraic groups who find they need to know something about Jordan algebras and will provide Jordan algebraists with new techniques and a new approach to finite-dimensional algebras over fields." (American Scientist) "By placing the classification of Jordan algebras in the perspective of classification of certain root systems, the book demonstrates that the structure theories associative, Lie, and Jordan algebras are not separate creations but rather instances of the one all-encompassing miracle of root systems. ..." (Math. Reviews)

• Area
• Finite
• Lie
• Math
• algebra
• algebraic group
• character
• classification
• geometry
• group
• lie algebra
• mathematics
• review
• service
• techniques

From the reviews: "This book presents an important and novel approach to Jordan algebras. Jordan algebras have come to play a role in many areas of mathematics, including Lie algebras and the geometry of Chevalley groups. Springer's work will be of service to research workers familiar with linear algebraic groups who find they need to know something about Jordan algebras and will provide Jordan algebraists with new techniques and a new approach to finite-dimensional algebras over fields." (American Scientist) "By placing the classification of Jordan algebras in the perspective of classification of certain root systems, the book demonstrates that the structure theories associative, Lie, and Jordan algebras are not separate creations but rather instances of the one all-encompassing miracle of root systems. ..." (Math. Reviews)

• Mathematics Department, University of Utrecht, Utrecht, The Netherlands

  Tonny A. Springer

Biography of Tonny A. Springer

Born on February 13, 1926 at the Hague, Holland, Tonny A. Springer studied mathematics at the University of Leiden, obtaining his Ph. D. in 1951. He has been at the University of Utrecht since 1955, from 1959-1991 as a full professor, and since 1991 as an emeritus professor.
He has held visiting positions at numerous prestigious institutions all over the globe, including the Institute for Advanced Study (Princeton), the Institut des Hautes Etudes Scientifiques (Bures-sur-Yvette) and the Tata Institute of Fundamental Research (Bombay).
Throughout his career T. A. Springer has been involved in research on various aspects of the theory of linear algebraic groups (conjugacy classes, Galois cohomology, Weyl groups).

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