Semisimple subalgebras of semisimple Lie algebras

Reflection subgroups of finite and affine Weyl groups

“…Given Corollary 2, Theorem 1 and Corollary 1 are easily deduced from the results of Dynkin [11] or Borel and de Siebenthal [21]. However, we do not know any independent simple proof of Corollary 2.…”

“…The closed root subsystems were determined up to isomorphism by Borel and de Siebenthal [1] (who treated the maximal rank ones) and by Dynkin [11], using the connection between crystallographic Weyl groups and Lie groups or Lie algebras. A treatment of these results directly in terms of root systems may be found in [21].…”

“…The paper [12] gives a case-by-case description of the reflection subgroups of finite Coxeter groups up to isomorphism (even up to automorphisms of W in the case of maximal reflection subgroups). In the case of Weyl groups, the results are based on the tables in Dynkin [11,Ch 2,§5], giving closed subsystems and maximal closed subsystems of Φ up to W -action. In the book [22] the classification of reflection subgroups up to isomorphism was extended from Weyl groups to all unitary (complex) reflection groups.…”

“…A Lie subalgebra g 1 of a complex semisimple or Kac-Moody Lie algebra g is said to be regular if it is invariant under a Cartan subalgebra h of g. The root system of a regular subalgebra g 1 with respect to h may naturally be regarded as a closed subsystem of that of g. The regular semisimple subalgebras of a simple complex Lie algebra were classified in [11,Ch. 2,§5] for example (hence the emphasis on closed subsystems there).…”

Orbits in a real reductive Lie algebra

“…Since x is the monosemisimple part of an ^-triple, it follows that n¡ -0, \, or 1 for each i. (For proof of this see Kostant [5, Lemma 5.1] or Dynkin [2].) Since all the irreducible components are odd-dimensional, it follows that n^\ for all /.…”