# Locally Finite Root Systems

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*Ottmar Loos; Erhard Neher*

#### Table of Contents

# Table of Contents

## Locally Finite Root Systems

- Contents vii8 free
- Introduction 112 free
- 1. The category of sets in vector spaces 617 free
- 2. Finiteness conditions and bases 1425
- 3. Locally finite root systems 2132
- 4. Invariant inner products and the coroot system 2839
- 5. Weyl groups 3849
- 6. Integral bases, root bases and Dynkin diagrams 4758
- 7. Weights and coweights 5364
- 8. Classification 6475
- 9. More on Weyl groups and automorphism groups 7586
- 10. Parabolic subsets and positive systems for symmetric sets in vector spaces 8596
- 11. Parabolic subsets of root systems and presentations of the root lattice and the Weyl group 97108
- 12. Closed and full subsystems of finite and infinite classical root systems 110121
- 13. Parabolic subsets of root systems: classification 128139
- 14. Positive systems in root systems 138149
- 15. Positive linear forms and facets 146157
- 16. Dominant and fundamental weights 153164
- 17. Gradings of root systems 165176
- 18. Elementary relations and graphs in 3-graded root systems 174185
- Appendix A. Some standard results on finite root systems 185196
- Appendix B. Cones defined by totally preordered sets 189200
- Bibliography 201212
- Index of notations 205216 free
- Index 211222