Borel–de Siebenthal theory

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 C_G(S), there is a maximal torus containing both S and x and it is contained in C_G(S).[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.[2]

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.[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 C_G(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.[4]

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 ${\displaystyle {\mathfrak {g}}}$ of a connected compact Lie group G splits as a direct sum of the ideals

${\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {z}}\oplus {\mathfrak {g}}_{1}\oplus \cdots \oplus {\mathfrak {g}}_{m},}}$

where ${\displaystyle {\mathfrak {z}}}$ is the center and the other factors ${\displaystyle {\mathfrak {g}}_{i}}$ are simple. If T is a maximal torus, its Lie algebra ${\displaystyle {\mathfrak {t}}}$ has a corresponding splitting

${\displaystyle \displaystyle {{\mathfrak {t}}={\mathfrak {z}}\oplus {\mathfrak {t}}_{1}\oplus \cdots \oplus {\mathfrak {t}}_{m},}}$

where ${\displaystyle {\mathfrak {t}}_{i}}$ is maximal abelian in ${\displaystyle {\mathfrak {g}}_{i}}$. If H is a closed connected of G containing T with Lie algebra ${\displaystyle {\mathfrak {h}}}$, the complexification of ${\displaystyle {\mathfrak {h}}}$ is the direct sum of the complexification of ${\displaystyle {\mathfrak {t}}}$ and a number of one-dimensional weight spaces, each of which lies in the complexification of a factor ${\displaystyle {\mathfrak {g}}_{i}}$. Thus if

${\displaystyle \displaystyle {{\mathfrak {h}}_{i}={\mathfrak {h}}\cap {\mathfrak {g}}_{i},}}$

then

${\displaystyle \displaystyle {{\mathfrak {h}}={\mathfrak {z}}\oplus {\mathfrak {h}}_{1}\oplus \cdots \oplus {\mathfrak {h}}_{m}.}}$

If H is maximal, all but one of the ${\displaystyle {\mathfrak {h}}_{i}}$'s coincide with ${\displaystyle {\mathfrak {g}}_{i}}$ 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.[5]

Let G be a connected simply connected compact simple Lie group with maximal torus T. Let ${\displaystyle {\mathfrak {g}}}$ be the Lie algebra of G and ${\displaystyle {\mathfrak {t}}}$ that of T. Let Δ be the corresponding root system. Choose a set of positive roots and corresponding simple roots α₁, ..., α_n. Let α₀ the highest root in ${\displaystyle {\mathfrak {g}}_{\mathbf {C} }}$ and write

${\displaystyle \displaystyle {\alpha _{0}=m_{1}\alpha _{1}+\cdots +m_{n}\alpha _{n}}}$

with m_i ≥ 1. (The number of m_i equal to 1 is equal to |Z| – 1, where Z is the center of G.)

The Weyl alcove is defined by

${\displaystyle \displaystyle {A=\{T\in {\mathfrak {t}}:\,\alpha _{1}(T)\geq 0,\dots ,\alpha _{n}(T)\geq 0,\alpha _{0}(T)\leq 1\}.}}$

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 ${\displaystyle {\mathfrak {g}}}$ to G₁ carries 2πA onto T₁.

The Weyl alcove A is a simplex with vertices at

${\displaystyle \displaystyle {v_{0}=0,\,\,v_{i}=m_{i}^{-1}X_{i},}}$

where α_i(X_j) = δ_ij.

The main result of Borel and de Siebenthal is as follows.

Extended Dynkin diagrams for the simple complex Lie algebras

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 α_i by −α₀. In the first case it is the direct product of the circle group generated by X_i and a semisimple compact group with a system of simple roots obtained by omitting α_i.

This result can be rephrased in terms of the extended Dynkin diagram of ${\displaystyle {\mathfrak {g}}}$ which adds an extra node for the highest root as well as the labels m_i. The maximal subalgebras ${\displaystyle {\mathfrak {h}}}$ of maximal rank are either non-semisimple or semisimple. 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 ${\displaystyle {\mathfrak {h}}}$, 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:

• A_n: A_p × A _{n − p − 1} × T (non-semisimple)
• B_n: D_n or B_p × D_{n − p} (semisimple), B_{n − 1} × T (non-semisimple)
• C_n: C_p × C_{n − p} (SS), A_{n - 1} × T (NSS)
• D_n: D_p × D_{n - p} (SS), D_{n - 1} × T, A_n-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 than that edge connects 0 to a vertex v_i with m_i = 1, which is the first case. If T is a vertex v_i and m_i has a non-trivial factor m, then mT has a larger stabilizer than T, contradicting maximality. So m_i must be prime. 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. If the center of H₁ is 'T, each simple root with m_i prime is already a root of K, so (b) is not possible; and if (a) holds, α_i is the only root that could be omitted with m_j = 1, so K = H₁. If the center of H₁ is of prime order, α_j is a root of K for m_j = 1, so that (a) is not possible; if (b) holds, then the only possible omitted simple root is α_i, so that K = H₁.[6]

A subset Δ₁ ⊂ Δ is called a closed subsystem if whenever α and β lie in Δ₁ with α + β in Δ, then α + β lies in Δ₁. Two subsystems Δ₁ and Δ₂ are said to be equivalent if σ( Δ₁) = Δ₂ for some σ in W = N_G(T) / T, the Weyl group. Thus for a closed subsystem

${\displaystyle \displaystyle {{\mathfrak {t}}_{\mathbf {C} }\oplus \bigoplus _{\alpha \in \Delta _{1}}{\mathfrak {g}}_{\alpha }}}$

is a subalgebra of ${\displaystyle {\mathfrak {g}}_{\mathbf {C} }}$ containing ${\displaystyle {\mathfrak {t}}_{\mathbf {C} }}$; and conversely any such subalgebra gives rise to a closed subsystem. Borel and de Siebenthal classified the maximal closed subsystems up to equivalence.[7]

This result is a consequence of the Borel–de Siebenthal theorem for maximal connected subgroups of maximal rank. It can also be proved directly within the theory of root systems and reflection groups.[8]

Let G be a connected compact semisimple Lie group, σ an automorphism of G of period 2 and G^σ the fixed point subgroup of σ. Let K be a closed subgroup of G lying between G^σ and its identity component. The compact homogeneous space G / K is called a symmetric space of compact type. The Lie algebra ${\displaystyle {\mathfrak {g}}}$ admits a decomposition

${\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}},}}$

where ${\displaystyle {\mathfrak {k}}}$, the Lie algebra of K, is the +1 eigenspace of σ and ${\displaystyle {\mathfrak {p}}}$ the –1 eigenspace. If ${\displaystyle {\mathfrak {k}}}$ contains no simple summand of ${\displaystyle {\mathfrak {g}}}$, the pair (${\displaystyle {\mathfrak {g}}}$, σ) is called an orthogonal symmetric Lie algebra of compact type.[9]

Any inner product on ${\displaystyle {\mathfrak {g}}}$, invariant under the adjoint representation and σ, induces a Riemannian structure on G / K, with G acting by isometries. Under such an inner product, ${\displaystyle {\mathfrak {k}}}$ and ${\displaystyle {\mathfrak {p}}}$ are orthogonal. G / K is then a Riemannian symmetric space of compact type.[10]

The symmetric space or the pair (${\displaystyle {\mathfrak {g}}}$, σ) is said to be irreducible if the adjoint action of ${\displaystyle {\mathfrak {k}}}$ (or equivalently the identity component of G^σ or K) is irreducible on ${\displaystyle {\mathfrak {p}}}$. This is equivalent to the maximality of ${\displaystyle {\mathfrak {k}}}$ as a subalgebra.[11]

In fact there is a one-one correspondence between intermediate subalgebras ${\displaystyle {\mathfrak {h}}}$ and K-invariant subspaces ${\displaystyle {\mathfrak {p}}_{1}}$ of ${\displaystyle {\mathfrak {p}}}$ given by

${\displaystyle \displaystyle {{\mathfrak {h}}={\mathfrak {k}}\oplus {\mathfrak {p}}_{1},\,\,\,\ {\mathfrak {p}}_{1}={\mathfrak {h}}\cap {\mathfrak {p}}.}}$

Any orthogonal symmetric algebra (${\displaystyle {\mathfrak {g}}}$, σ) can be decomposed as an (orthogonal) direct sum of irreducible orthogonal symmetric algebras.[12]

In fact ${\displaystyle {\mathfrak {g}}}$ can be written as a direct sum of simple algebras

${\displaystyle \displaystyle {{\mathfrak {g}}=\oplus _{i=1}^{N}{\mathfrak {g}}_{i},}}$

which are permuted by the automorphism σ. If σ leaves an algebra ${\displaystyle {\mathfrak {g}}_{1}}$ invariant, its eigenspace decomposition coincides with its intersections with ${\displaystyle {\mathfrak {k}}}$ and ${\displaystyle {\mathfrak {p}}}$. So the restriction of σ to ${\displaystyle {\mathfrak {g}}_{1}}$ is irreducible. If σ interchanges two simple summands, the corresponding pair is isomorphic to a diagonal inclusion of K in K × K, with K simple, so is also irreducible. The involution σ just swaps the two factors σ(x,y)=(y,x).

This decomposition of an orthogonal symmetric algebra yields a direct product decomposition of the corresponding compact symmetric space G / K when G is simply connected. In this case the fixed point subgroup G^σ is automatically connected (this is no longer true, even for inner involutions, if G is not simply connected).[13] For simply connected G, the symmetric space G / K is the direct product of the two kinds of symmetric spaces G_i / K_i or H × H / H. Non-simply connected symmetric space of compact type arise as quotients of the simply connected space G / K by finite abelian groups. In fact if

${\displaystyle \displaystyle {G/K=G_{1}/K_{1}\times \cdots \times G_{s}/K_{s},}}$

let

${\displaystyle \displaystyle {\Gamma _{i}=Z(G_{i})/Z(G_{i})\cap K_{i}}}$

and let Δ_i be the subgroup of Γ_i fixed by all automorphisms of G_i preserving K_i (i.e. automorphisms of the orthogonal symmetric Lie algebra). Then

${\displaystyle \displaystyle {\Delta =\Delta _{1}\times \cdots \times \Delta _{s}}}$

is a finite abelian group acting freely on G / K. The non-simply connected symmetric spaces arise as quotients by subgroups of Δ. The subgroup can be identified with the fundamental group, which is thus a finite abelian group.[14]

The classification of compact symmetric spaces or pairs (${\displaystyle {\mathfrak {g}}}$, σ) thus reduces to the case where G is a connected simple compact Lie group. There are two possibilities: either the automorphism σ is inner, in which case K has maximal rank and the theory of Borel and de Siebenthal applies; or the automorphism σ is outer, so that, because σ preserves a maximal torus, the rank of K is less than the rank of G and σ corresponds to an automorphism of the Dynkin diagram modulo inner automorphisms. Wolf (2010) determines directly all possible σ in the latter case: they correspond to the symmetric spaces SU(n)/SO(n), SU(2n)/Sp(n), SO(a+b)/SO(a)×SO(b) (a and b odd), E₆/F₄ and E₆/C₄.[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 (${\displaystyle {\mathfrak {g}}}$, σ), is described in Helgason (1978).

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 ${\displaystyle {\mathfrak {p}}}$ which preserves the inner product and commutes with the action of K. In this case J lies in ${\displaystyle {\mathfrak {k}}}$ and exp Jt forms a one-parameter group in the center of K. This follows because if A, B, C, D lie in ${\displaystyle {\mathfrak {p}}}$, then by the invariance of the inner product on ${\displaystyle {\mathfrak {g}}}$[16]

${\displaystyle \displaystyle {([[A,B],C],D)=([A,B],[C,D])=([[C,D],B],A).}}$

Replacing A and B by JA and JB, it follows that

${\displaystyle \displaystyle {[JA,JB]=[A,B].}}$

Define a linear map δ on ${\displaystyle {\mathfrak {g}}}$ by extending J to be 0 on ${\displaystyle {\mathfrak {k}}}$. The last relation shows that δ is a derivation of ${\displaystyle {\mathfrak {g}}}$. Since ${\displaystyle {\mathfrak {g}}}$ is semisimple, δ must be an inner derivation, so that

${\displaystyle \displaystyle {\delta (X)=[T+A,X],}}$

with T in ${\displaystyle {\mathfrak {k}}}$ and A in ${\displaystyle {\mathfrak {p}}}$. Taking X in ${\displaystyle {\mathfrak {k}}}$, it follows that A = 0 and T lies in the center of ${\displaystyle {\mathfrak {k}}}$ and hence that K is non-semisimple. [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 G_C of G such that G / K = G_C / 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 G_i / K_i with G_i simple. The irreducible ones are exactly the non-semisimple cases described above.[18]

• Borel, A.; De Siebenthal, J. (1949), "Les sous-groupes fermés de rang maximum des groupes de Lie clos", Commentarii Mathematici Helvetici, 23: 200–221, doi:10.1007/bf02565599, S2CID 120101481
• Borel, Armand (1952), Les espaces hermitiens symétriques, Exposé No. 62, Séminaire Bourbaki, vol. 2, archived from the original on 2016-03-04, retrieved 2013-03-14
• Bourbaki, N. (1981), Groupes et Algèbres de Lie (Chapitres 4,5 et 6), Éléments de Mathématique, Masson, ISBN 978-3-540-34490-2
• Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN 978-3-540-34392-9
• Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 978-3-540-15293-4
• Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 978-0-8218-2848-9
• Humphreys, James E. (1981), Linear Algebraic Groups, Graduate texts in mathematics, vol. 21, Springer, ISBN 978-0-387-90108-4
• Humphreys, James E. (1997), Introduction to Lie Algebras and Representation Theory, Graduate texts in mathematics, vol. 9 (2nd ed.), Springer, ISBN 978-3-540-90053-5
• Kane, Richard (2001), Reflection Groups and Invariant Theory, Springer, ISBN 978-0-387-98979-2
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996), Foundations of differential geometry, vol. 2, Wiley-Interscience, ISBN 978-0-471-15732-8
• Malle, Gunter; Testerman, Donna (2011), Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, ISBN 978-1-139-49953-8
• Wolf, Joseph A. (2010), Spaces of constant curvature, AMS Chelsea Publishing (6th ed.), American Mathematical Society, ISBN 978-0-8218-5282-8