Contents
Introduction 1
1. The category of sets in vector spaces 6
2. Finiteness conditions and bases 14
3. Locally finite root systems 21
4. Invariant inner products and the coroot system 28
5. Weyl groups 38
6. Integral bases, root bases and Dynkin diagrams 47
7. Weights and coweights 53
8. Classification 64
9. More on Weyl groups and automorphism groups 75
10. Parabolic subsets and positive systems
for symmetric sets in vector spaces 85
11. Parabolic subsets of root systems
and presentations of the root lattice and the Weyl group 97
12. Closed and full subsystems of finite and infinite
classical root systems 110
13. Parabolic subsets of root systems: classification 128
14. Positive systems in root systems 138
15. Positive linear forms and facets 146
16. Dominant and fundamental weights 153
17. Gradings of root systems 165
18. Elementary relations and graphs in 3-graded root systems 174
A. Some standard results on finite root systems 185
B. Cones defined by totally preordered sets 189
Bibliography 201
Index of notations 205
Index 211
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