Einstein Equation for an Invariant Metric on Generalized Flag Manifolds and Inner Automorphisms

Andreas Arvanitoyeorgos

Abstract A generalized flag manifold is a homogeneous space G/K whose isotropy subgroup K is the centralizer of a torus in G. We show that the Einstein equation for a G-invariant metric on G/K is invariant under the group of inner automorphisms of the Lie algebra of G that preserve the Lie algebra of K and a fixed Cartan subalgebra of K.

Mathematics Subject Classification: 53C25, 53C30, 17B05 Key words: Einstein metric, flag manifold, inner automorphism

1 Introduction

A metric g on a Riemannian manifold M is called an Einstein metric if Ric(g) = cg, where Ric(g) is the Ricci tensor of the metric g, and c is a constant. In [1] we presented new G-invariant Einstein metrics for certain homogeneous spaces G/K called generalized flag manifolds. These metrics were non-Kähler and different from the normal metric [9]. The methodology we used there was the reduction of the Einstein equation to an algebraic system of equations through a Lie theoretic description of the Ricci curvature Ric(g) and the G-invariant metric g. Let g and k be the Lie algebras of G and K respectively, and h a fixed Cartan subalgebra of k. In this paper we show that the Einstein equation for a generalized flag manifold is invariant under the group of inner automorphisms of g that preserve h and k.

2 Generalized flag manifolds: Lie theoretic description

Let G be a compact, connected and semisimple Lie group. A generalized flag manifold is a homogeneous space M = G/K whose isotropy group K is the centralizer of a torus in G. Equivalently, M is the adjoint orbit Ad(G)w (w some element in g) of w under the action of the adjoint representation Ad of G in g [5, 8]. Since G is semisimple and compact, the Killing form B(X, Y) = trad(X)ad(Y) of g is nondegenerate and negative definite on g, thus giving rise to an orthogonal decomposition of g as the direct sum g = k ⊕ m. This sum is Ad(K)-invariant, i.e. [m, k] ⊂ m. Moreover the tangent space T_{p}M of M can be identified with m. This identification is given by

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 17-22 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

X ↦ X^{∗}(p) = \frac{d}{dt}(exp tX · p)|_{t=0} , X ∈ m, p ∈ M.

We fix a Cartan subalgebra hC of the complexified Lie algebra kC; then the Cartan decompositions of gC and kC are given as follows:

gC = hC + ∑_{α∈R} g^{(α)}, kC = hC + ∑_{α∈R_{K}} g^{(α)}, mC = ∑_{α∈R_{M}} g^{(α)},

where R and R_{K} are the root systems of the pairs (gC, hC) and (kC, hC) respectively. The root system R is decomposed as R = R_{K} ∪ R_{M} where R_{M} is the set of complementary roots. The spaces g^{(α)} are the 1-dimensional root spaces whose elements X_{α} are characterized by the equation [H, X_{α}] = α(H)X_{α}, H ∈ hC. We also recall that for any root α we can choose elements E_{α} ∈ g^{(α)} (α ∈ R) which have the properties B(E_{α}, E_{−α}) = −1, [E_{α}, E_{−α}] = −H_{α}, where H_{α} is determined by the equation B(H_{α}, H) = α(H) for all H ∈ hC, as well as [E_{α}, E_{β}] = N_{αβ}E_{α+β} for α, β ∈ R, α + β ∈ R, with coefficients N_{α,β} (structural constants). The set {E_{α} : α ∈ R_{M}} constitutes a basis for mC.

3 Invariant metrics

Having the decomposition gC = kC ⊕ mC associated with the generalized flag manifold M = G/K and the decomposition R = R_{K} ∪ R_{M} of the root system R, we set h = g ∩ hC and define

t = Z(kC) ∩ h = {X ∈ h : ϕ(X) = 0 ∀ϕ ∈ R_{K}}.

If h^{∗} and t^{∗} are the dual spaces of h and t, consider the restriction map

\begin{matrix}κ : h^{∗} → t^{∗} \\ α ↦ κ(α) = α|t \end{matrix}

and set Rt ≡ κ(R) = κ(R_{M}) (note that κ(R_{K}) = 0). The elements of Rt are called t-roots. The benefit from these is that there is a one-to-one correspondence between t-roots τ and irreducible adg(kC)-invariant submodules MC_{τ} = ∑_{κ(α)=τ} CE_{α} of mC

[6, 2].

Now, a G-invariant metric on M = G/K can be described by an adg(k)-invariant scalar product g on m, and we extend without any change in notation to mC. Let {ω^{α} : α ∈ R} be the vector space basis in (mC)^{∗} which is dual to the basis {E_{α} : α ∈ R_{M}}. We fix a system of positive roots R^{+} = R_{K}^{+} ∪R_{M}^{+}, where R_{K}^{+} = R_{K} ∩R^{+}, R_{M}^{+} = R_{M} ∩ R^{+}, and set Rt^{+} = κ(R^{+}). The following proposition gives a description of the invariant metrics on M.

Proposition 1 [3].Any real adg(kC)-invariant scalar product g on mC has the form

g = ∑_{α∈R_{M}^{+}} g_{α}ω^{α} ∨ ω^{−α} = ∑_{τ∈Rt^{+}} g_{τ} ∑_{α∈κ^{−1}(τ)} ω^{α} ∨ ω^{−α},

where ω ∨ ρ = \frac{1}{2}(ω ⊗ ρ + ρ ⊗ ω), g_{α} ∈ R^{+}, and g_{α} = g_{β} if α|t = β|t so the invariant Riemannian metrics on a generalized flag manifold M = G/K depend (modulo the scale factor) on |Rt^{+}| parameters.

4 The Ricci tensor and the Einstein equation

The Ricci tensor can now be determined by its value on the basis {E_{α} : α ∈ R_{M}}. We have the following

Proposition 2 [2]. The Ricci tensor for an invariant metric g described in proposition 1 is given by

Ric(E_{α}, E_{β}) = 0 , α, β ∈ R_{M} , α + β ∉ R_{M}

Ric(E_{α}, E_{−α}) = (α, α) + ∑_{{ϕ∈R_{K} \atop α+ϕ∈R}} N_{α,ϕ}^{2} + \frac{1}{4} ∑_{β∈R_{M}^{∗}} \frac{N_{α,β}^{2}}{g_{α+β}g_{β}} (g_{α}^{2} − (g_{α+β} − g_{β}^{2}),

where R_{M}^{∗} = R_{M} − κ^{−1}(κ(α)). Thus the Einstein equation Ric(g) = cg reduces (after normalizing either one of the g_{α} or c to 1) to an algebraic system of |Rt^{+}| equations with |Rt^{+}| unknowns.

5 Inner automorphisms

We recall that Adz, z ∈ G is the derivative of the conjugation C_{z} : g ↦ zgz^{−1} in G. The group of inner automorphisms of a complex Lie algebra g consists of finite products of the form Adz, z ∈ G, and it is a subgroup of the group of all automorphisms of g. Further, the Weyl group of the root system R is the set of all linear transformations on hR = ∑_{α∈R} RH_{α} induced by inner automorphisms of g that preserves h. We can

now state the main theorem.

Theorem.The set of equations that determine the Einstein condition for the generalized flag manifold M = G/K as given in proposition 2 is invariant under the group of inner automorphisms of g that preserve h and k. Equivalently, it is invariant under those elements in the Weyl group of R that preserve R_{K}.

Proof. Without loss of generality we need to examine the effect of w = Adz on the root elements E_{α}, the structural constants N_{α,β}, the components g_{α} of the G-invariant metric g, and finally on the set of equations that determine the Einstein equation.

STEP 1 Action of Adz on E_{α}.

Since Adz preserves h the equation α^{∗}(H) = B(H, Adz(H_{α})) defines a root α^{∗} so that Adz(H_{α}) = H_{α^{∗}}.

Claim: Adz(E_{α}) = E_{α^{∗}}.

We apply Adz to the equation [H, E_{α}] = α(H)E_{α} and obtain

(1)  [Adz(H), Adz(E_{α})] = α(H)Adz(E_{α}).

By the invariance of the Killing form under Adz we have that

(2)

α(H) = B(H, H_{α}) = B(Adz(H), Adz(H_{α}))
= B(Adz(H), H_{α^{∗}}) = α^{∗}(Adz(H)).

From (1) and (2) we obtain

[Adz(H), Adz(E_{α})] = α^{∗}(Adz(H))Adz(E_{α}),

which implies that Adz(E_{α}) is the root vector E_{α^{∗}} corresponding to the root α^{∗} up to a constant. However the E_{α}’s have been chosen so that this constant is normalized to 1. Notice that α^{∗} also satisfies the equation

α^{∗}(H) = B(Adz^{−1}(H), H_{α}) = α(Adz^{−1}(H)) = w · α(H)

which is the usual definition of the action of an element w in the Weyl group on the roots.

STEP 2 Transformation of N_{α,β}.

The numbers N_{α,β} are determined by the equation [E_{α}, E_{α}] = N_{α,β}E_{α+β} (α + β ∈ R). Applying Adz to this equation and using step 1 we get [E_{α^{∗}}, E_{β^{∗}}] = N_{α,β}E_{(α+β)^{∗}} (α^{∗} + β^{∗} ∈ R), or N_{α^{∗},β^{∗}}E_{α^{∗}+β^{∗}} = N_{α,β}E_{(α+β)^{∗}}. The last equation determines the action of Adz on N_{α,β} implicitly.

STEP 3 Transformation of g_{α}.

Let w be an element in the Weyl group of R that preserves R_{K}. Then the diffeomorphism C_{z} preserves K, thus it induces a map \overline{C}_{z} on G/K. Then Adz restricts to a map d\overline{C}_{z} on m = T_{o}(G/K), and consequently it takes an invariant metric g to a new invariant metric Adz · g defined by

Adz · g(X, Y) = g(Adz(X), Adz(Y)).

For X = E_{α} and Y = E_{−α} this gives

Adz · g_{α} = g(E_{α^{∗}}, E_{(−α)^{∗}}) = g_{α^{∗}}.

STEP 4 Transformation of the system of equations.

We apply w = AdZ to the system of equations in proposition 2 and obtain

(3) (α^{∗}, α^{∗}) = ∑_{{ϕ^{∗}∈R_{K} \atop α^{∗}+ϕ^{∗}∈R}} N_{α^{∗},ϕ^{∗}}^{2} + \frac{1}{4} ∑_{β^{∗}∈R_{M}^{∗}} \frac{N_{α^{∗},β^{∗}}^{2}}{g_{α^{∗}+β^{∗}}g_{β^{∗}}} (g_{α^{∗}}^{2} − (g_{α^{∗}+β^{∗}} − g_{β^{∗}})^{2}) = g_{α^{∗}}.

We need to show that (3) is equivalent to

(4) (α^{∗}, α^{∗}) = ∑_{{ϕ∈R_{K} \atop al^{∗}+ϕ∈R}} N_{α^{∗},ϕ}^{2} + \frac{1}{4} ∑_{β∈R_{M}^{∗}} \frac{N_{α^{∗},β}^{2}}{g_{α^{∗}+β}g_{β}} (g_{α^{∗}}^{2} − (g_{α^{∗}+β} − g_{β})^{2}) = g_{α^{∗}}.

Since α^{∗} = w · α we can replace ϕ and β in (4) by ψ^{∗} and γ^{∗} respectively (for some ψ and γ ∈ R). Then we can use the invariance of R_{K} under w to obtain equation (3).

Example. Let G/K = SU(n)/S(U(n_{1}) × ⋯ × U(n_{s})), n = ∑_{i=1}^{s} n_{i}.

According to [3] and [1] the Einstein equation reduces to the following system

n_{i} + n_{j} + \frac{1}{2} ∑_{l≠i,j} \frac{n_{l}}{g_{il}g_{jl}} (g_{ij}^{2} − (g_{il} − g_{jl})^{2}) = g_{ij}

of \frac{1}{2}s(s − 1) equations with \frac{1}{2}s(s − 1) unknowns g_{ij}, the components of the SU(n)- invariant metric g. The Weyl group of SU(n) is the group of permutations w of the set {1, …, n} which acts on the set R = {ϵ_{i} − ϵ_{j} : i ≠ j} of roots of SU(n) according to w(ϵ_{i} − ϵ_{j}) = ϵ_{w(i)} − ϵ_{w(j)}, and on g_{ij} by wg_{ij} = g_{w(i),w(j)}. The integers n_{i} are transformed by wn_{i} = n_{w(i)}. Since w preserves R_{K} the set of equivalence classes {[g_{ij}] : g_{ij} ∼ g_{kl} ⇔ κ(ϵ_{i} − ϵ_{j}) = κ(ϵ_{k} − ϵ_{l})} is closed under w. Thus the equations are preserved.

References

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Andreas Arvanitoyeorgos Filis 181 GR-11252 Athens, Greece aarv@leon.nrcps.oriadne-t.gr