page_title: Borel–de Siebenthal theory
section_title: Maximal connected subgroups of maximal rank
hierarchical_section_title: Borel–de Siebenthal theory / Maximal connected subgroups of maximal rank
caption_attribution_description: English: Affine Dynkin diagrams with standard labelling, produced in TeX
context_page_description: 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.
context_section_description: Borel and de Siebenthal classified the maximal closed connected subgroups of maximal rank of a connected compact Lie group.
  The general classification of connected closed subgroups of maximal rank can be reduced to this case, because any connected subgroup of maximal rank is contained in a finite chain of such subgroups, each maximal in the next one. Maximal subgroups are the identity components of any element of their center not belonging to the center of the whole group.
  The problem of determining the maximal connected subgroups of maximal rank can be further reduced to the case where the compact Lie group is simple. In fact the Lie algebra  of a connected compact Lie group G splits as a direct sum of the ideals
  where  is the center and the other factors  are simple. If T is a maximal torus, its Lie algebra  has a corresponding splitting
  where  is maximal abelian in . If H is a closed connected of G containing T with Lie algebra , the complexification of  is the direct sum of the complexification of  and a number of one-dimensional weight spaces, each of which lies in the complexification of a factor . Thus if
  then
  If H is maximal, all but one of the 's coincide with  and the remaining one is maximal and of maximal rank. For that factor, the closed connected subgroup of the corresponding simply connected simple compact Lie group is maximal and of maximal rank.
  Let G be a connected simply connected compact simple Lie group with maximal torus T. Let  be the Lie algebra of G and  that of T. Let Δ be the corresponding root system. Choose a set of positive roots and corresponding simple roots α₁, ..., αₙ. Let α₀ the highest root in  and write
  with mᵢ ≥ 1. (The number of mᵢ equal to 1 is equal to |Z| – 1, where Z is the center of G.)
  The Weyl alcove is defined by
  Élie Cartan showed that it is a fundamental domain for the affine Weyl group. If G₁ = G / Z and T₁ = T / Z, it follows that the exponential mapping from  to G₁ carries 2πA onto T₁.
  The Weyl alcove A is a simplex with vertices at
  where αᵢ(Xⱼ) = δᵢⱼ.
  The main result of Borel and de Siebenthal is as follows.
  The structure of the corresponding subgroup H₁ can be described in both cases. It is semisimple in the second case with a system of simple roots obtained by replacing αᵢ by −α₀. In the first case it is the direct product of the circle group generated by Xᵢ and a semisimple compact group with a system of simple roots obtained by omitting αᵢ.
  This result can be rephrased in terms of the extended Dynkin diagram of  which adds an extra node for the highest root as well as the labels mᵢ. The maximal subalgebras  of maximal rank are either non-semisimple or semimsimple. The non-semisimple ones are obtained by deleting two nodes from the extended diagram with coefficient one. The corresponding unlabelled diagram gives the Dynkin diagram semisimple part of , the other part being a one-dimensional factor.  The Dynkin diagrams for the semisimple ones are obtained by removing one node with coefficient a prime. This leads to the following possibilities:
  Aₙ: Aₚ × A ₙ − ₚ − 1 × T (non-semisimple)
  Bₙ: Dₙ or Bₚ × Dₙ − ₚ (semisimple), Bₙ − 1 × T (non-semisimple)
  Cₙ: Cₚ × Cₙ − ₚ (SS), Aₙ ₋ ₁ × T (NSS)
  Dₙ: Dₚ × Dₙ ₋ ₚ (SS), Dₙ ₋ ₁ × T, Aₙ₋₁ × T (NSS)
  E₆: A₁ × A₅, A₂ × A₂ × A₂ (SS), D₅ × T (NSS)
  E₇: A₁ × D₆, A₂ × A₅, A₇ (SS), E₆ × T (NSS)
  E₈: D₈, A₈, A₄ × A₄, E₆ × A₂, E₇ × A₁ (SS)
  F₄: B₄, A₂ × A₂, A₁ × C₃ (SS)
  G₂: A₂, A₁ × A₁ (SS)
  All the corresponding homogeneous spaces are symmetric, since the subalgebra is the fixed point algebra of an inner automorphism of period 2, apart from G₂/A₂, F₄/A₂×A₂, E₆/A₂×A₂×A₂, E₇/A₂×A₅ and all the E₈ spaces other than E₈/D₈ and E₈/E₇×A₁. In all these exceptional cases the subalgebra is the fixed point algebra of an inner automorphism of period 3, except for E₈/A₄×A₄ where the automorphism has period 5. 
  To prove the theorem, note that H₁ is the identity component of the centralizer of an element exp T with T in 2π A. Stabilizers increase in moving from a subsimplex to an edge or vertex, so T either lies on an edge or is a vertex. If it lies on an edge
caption: Extended Dynkin diagrams for the simple complex Lie algebras
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