Borel–de Siebenthal theory
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
Let G be connected compact Lie group with maximal torus T. Hopf showed that the centralizer of a torus S ⊆ T is a connected closed subgroup containing T, so of maximal rank. Indeed, if x is in CG(S), there is a maximal torus containing both S and x and it is contained in CG(S).
Borel and de Siebenthal proved that the connected closed subgroups of maximal rank are precisely the identity components of the centralizers of their centers.
Their result relies on a fact from representation theory. The weights of an irreducible representation of a connected compact semisimple group K with highest weight λ can be easily described (without their multiplicities): they are precisely the saturation under the Weyl group of the dominant weights obtained by subtracting off a sum of simple roots from λ. In particular, if the irreducible representation is trivial on the center of K (a finite abelian group), 0 is a weight.
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