Restricted root system

Group theory → Lie groups Lie groups
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
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Homogeneous spaces Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system
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Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Table of Lie groups

In mathematics, restricted root systems, sometimes called relative root systems, are the root systems associated with a symmetric space. The associated finite reflection group is called the restricted Weyl group. The restricted root system of a symmetric space and its dual can be identified. For symmetric spaces of noncompact type arising as homogeneous spaces of a semisimple Lie group, the restricted root system and its Weyl group are related to the Iwasawa decomposition of the Lie group.

See also

• Satake diagram

References

• Bump, Daniel (2004), Lie groups, Graduate Texts in Mathematics, 225, Springer, ISBN 0387211543 
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0821828487 
• Onishchik, A. L.; Vinberg, E. B. (1994), Lie Groups and Lie Algebras III: Structure of Lie Groups and Lie Algebras, Encyclopaedia of Mathematical Sciences, 41, Springer, ISBN 9783540546832 
• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathematical Society, ISBN 0821852825 
• Wolf, Joseph A. (1972), "Fine structure of Hermitian symmetric spaces", in Boothby, William; Weiss, Guido, Symmetric spaces (Short Courses, Washington University), Pure and Applied Mathematics, 8, Dekker, pp. 271–357