Steinberg Groups for Jordan Pairs (Progress in Mathematics)

The present monograph develops a unified theory of Steinberg groups, independent of matrix representations, based on the theory of Jordan pairs and the theory of 3-graded locally finite root systems. The development of this approach occurs over six chapters, progressing from groups with commutator relations and their Steinberg groups, then on to Jordan pairs, 3-graded locally finite root systems, and groups associated with Jordan pairs graded by root systems, before exploring the volume's main focus: the definition of the Steinberg group of a root graded Jordan pair by a small set of relations, and its central closedness. Several original concepts, such as the notions of Jordan graphs and Weyl elements, provide readers with the necessary tools from combinatorics and group theory. __Steinberg Groups for Jordan Pairs__ is ideal for PhD students and researchers in the fields of elementary groups, Steinberg groups, Jordan algebras, and Jordan pairs. By adopting a unified approach, anybody interested in this area who seeks an alternative to case-by-case arguments and explicit matrix calculations will find this book essential. Contents Preface Notation and Conventions CHAPTER I: GROUPS WITH COMMUTATOR RELATIONS §1. Nilpotent sets of roots §2. Reflection systems and root systems §3. Groups with commutator relations §4. Categories of groups with commutator relations §5. Weyl elements CHAPTER II: GROUPS ASSOCIATED WITH JORDAN PAIRS §6. Introduction to Jordan pairs §7. The projective elementary group I §8. The projective elementary group II §9. Groups over Jordan pairs CHAPTER III: STEINBERG GROUPS FOR PEIRCE GRADED JORDAN PAIRS §10. Peirce gradings §11. Groups defined by Peirce gradings §12. Weyl elements for idempotent Peirce gradings §13. Groups defined by sets of idempotents CHAPTER IV: JORDAN GRAPHS §14. 3-graded root systems §15. Jordan graphs and 3-graded root systems §16. Local structure §17. Classification of arrows and vertices §18. Bases §19. Triangles CHAPTER V: STEINBERG GROUPS FOR ROOT GRADED JORDAN PAIRS §20. Root gradings §21. Groups defined by root gradings §22. The Steinberg group of a root graded Jordan pair §23. Cogs §24. Weyl elements for idempotent root gradings §25. The monomial group §26. Centrality results CHAPTER VI: CENTRAL CLOSEDNESS §27. Statement of the main result and outline of the proof §28. Invariant alternating maps §29. Vanishing of the binary symbols §30. Vanishing of the ternary symbols §31. Definition of the partial sections §32. Proof of the relations Bibliography Subject Index Notation Index