Restricted root system

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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 restrcted 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[edit]


  • 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