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)
Simple Lie groups List of simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop
Lie algebras Exponential map Adjoint representation (group algebra) Killing form Index Lie point symmetry Simple Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Homogeneous spaces Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system
Representation theory Lie group representation Lie algebra representation
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
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