Borel–de Siebenthal theory
It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949.
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.
[1] Borel and de Siebenthal proved that the connected closed subgroups of maximal rank are precisely the identity components of the centralizers of their centers.
In particular, if the irreducible representation is trivial on the center of K (a finite abelian group), 0 is a weight.
[3] To prove the characterization of Borel and de Siebenthal, let H be a closed connected subgroup of G containing T with center Z.
The identity component L of CG(Z) contains H. If it were strictly larger, the restriction of the adjoint representation of L to H would be trivial on Z.
Any irreducible summand, orthogonal to the Lie algebra of H, would provide non-zero weight zero vectors for T / Z ⊆ H / Z, contradicting the maximality of the torus T / Z in L / Z.
of a connected compact Lie group G splits as a direct sum of the ideals where
The Weyl alcove A is a simplex with vertices at where αi(Xj) = δij.
It is semisimple in the second case with a system of simple roots obtained by replacing αi by −α0.
which adds an extra node for the highest root as well as the labels mi.
The non-semisimple ones are obtained by deleting two nodes from the extended diagram with coefficient one.
The Dynkin diagrams for the semisimple ones are obtained by removing one node with coefficient a prime.
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 T is a vertex vi and mi has a non-trivial factor m, then mT has a larger stabilizer than T, contradicting maximality.
Maximality can be checked directly using the fact that an intermediate subgroup K would have the same form, so that its center would be either (a) T or (b) an element of prime order.
Borel and de Siebenthal classified the maximal closed subsystems up to equivalence.
, σ) is called an orthogonal symmetric Lie algebra of compact type.
, invariant under the adjoint representation and σ, induces a Riemannian structure on G / K, with G acting by isometries.
can be written as a direct sum of simple algebras which are permuted by the automorphism σ.
Then is a finite abelian group acting freely on G / K. The non-simply connected symmetric spaces arise as quotients by subgroups of Δ.
, σ) thus reduces to the case where G is a connected simple compact Lie group.
[15] Victor Kac noticed that all finite order automorphisms of a simple Lie algebra can be determined using the corresponding affine Lie algebra: that classification, which leads to an alternative method of classifying pairs (
The equal rank case with K non-semisimple corresponds exactly to the Hermitian symmetric spaces G / K of compact type.
In fact the symmetric space has an almost complex structure preserving the Riemannian metric if and only if there is a linear map J with J2 = −I on
and exp Jt forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in
[16] Replacing A and B by JA and JB, it follows that Define a linear map δ on
[17] If on the other hand G / K is irreducible with K non-semisimple, the compact group G must be simple and K of maximal rank.
From the theorem of Borel and de Siebenthal, the involution σ is inner and K is the centralizer of a torus S. It follows that G / K is simply connected and there is a parabolic subgroup P in the complexification GC of G such that G / K = GC / P. In particular there is a complex structure on G / K and the action of G is holomorphic.
In general any compact hermitian symmetric space is simply connected and can be written as a direct product of irreducible hermitian symmetric spaces Gi / Ki with Gi simple.
