Subregular subalgebras and invariant generalized complex structures on Lie groups

0.14 MB·

by  Evgeny Mayanskiy

Subregular subalgebras and invariant generalized complex
structures on Lie groups
Evgeny Mayanskiy
7
January 3, 2017
1
0
2
Abstract
n
a We introduce the notion of a subregular subalgebra, which we believe is useful for classification of
J
subalgebrasofLiealgebras. Weuseittoconstructanon-regularinvariantgeneralizedcomplexstructure

1 on a Lie group. Asan illustration of thestudy of invariant generalized complex structures, we compute
them all for thereal forms of G2.
]
G
A 1 Subregular subalgebras
.
h
t Let g be a finite-dimensional complex Lie algebra, k 0 an integer.
a ≥
m Definition 1.1. A subalgebra s g is called subregular in codimension k if s is normalized by a codimension
⊂
[ k subalgebra of a Cartan subalgebra of g.

If k 1, then s g is called subregular strictly in codimension k if s g is subregular in codimension k,
1 ≥ ⊂ ⊂
but is not subregular in codimension k 1.
v
−
2
Notethatanysubalgebras gissubregular(strictly)incodimensionk forsomek 0,1,...,rank(g) .
2
⊂ ∈{ }
2 Regular subalgebras, as defined in [2], [4], are precisely those which are subregular in codimension 0.
0
0 This notion may be useful for an explicit classification of subalgebras of Lie algebras as in [10]. In this
.
1 note, we demonstrate how it can be applied to construction of invariant generalized complex structures on
0 Lie groups.
7
1
: 2 Invariant generalized complex structures
v
i
X
Invariant generalized complex structures on homogeneous spaces were studied in [11] and [1]. In particular,
r Alekseevsky, David and Milburn classified invariant generalized complex structures on Lie groups in terms
a
of the so-called admissible pairs. We will review a part of their classification.
Throughout this section, G denotes a finite-dimensional connected real Lie group, g the (real) Lie
0 0
algebra of G0, g = g0 R C its complexification and τ: g g the corresponding antiinvolution. If g is
⊗ →
semisimple and h0 g0 is a Cartan subalgebra, then h = h0 R C g denotes its complexification and
⊂ ⊗ ⊂
Φ h∗ the root system of g with respect to the Cartan subalgebra h.
⊂
Definition 2.1 (Alekseevsky-David [1], Milburn [11]). A g -admissible pair is a pair (s,ω), where s g is
0
⊂
a complex subalgebra and ω 2s∗ is a closed 2-form such that:
∈V
s+τ(s)=g, and
•
Im(ω ) is non-degenerate.
• |g0∩s
Theorem 2.2 (Akelseevsky-David [1], Milburn [11]). There is a one-to-one correspondence between the
invariant generalized complex structures on G and the g -admissible pairs (s,ω).
0 0
1
The following notion was introduced by Alekseevsky and David [1].
Definition 2.3 (Akelseevsky-David[1]). An invariant generalized complex structure on G is called regular
0
if the associated subalgebra s g is normalized by a Cartan subalgebra of g .
0
⊂
The following theorem strengthens [1], Theorem15 and completes the classificationof invariant general-
ized complex structures on finite-dimensional compact connected real semisimple Lie groups.
Theorem2.4. IfG isafinite-dimensionalcompactconnectedrealsemisimpleLiegroup,thenanyinvariant
0
generalized complex structure on G is regular.
0
Proof. Let s g be the complex subalgebra associatedby Theorem 2.2 to an invariant generalized complex
⊂
structure on G . Let N(s) g be its normalizer.
0
⊂
By [9], Theorem 13, N(s) g generates a closed subgroup of G . The same argument as in [1], Theo-
0 0
∩
rem 15, using [14], implies that N(s) is normalized by a Cartan subalgebra h g , i.e.
0 0
⊂
N(s)=L CX ,
⊕ M α
α∈R
where R Φ is a closed subset, X , α Φ, are root vectors of g with respect to the Cartan subalgebra
α
⊂ ∈
h=h0 RC, and L h is the solution set of a system of equations of the form α β =0, α,β Φ.
⊗ ⊂ − ∈
Since τ is the conjugation with respect to a compact real form g of g and τ(h)=h,
0
τ h(R)= Idh(R), τ(CXα)=CX−α,
| −
where h(R) is the real span in h of the coroots of g with respect to h [6].
Since N(s)+τ(N(s))=g, L+τ(L)=h, which is possible only if L=h. Hence h N(s) normalizes s.
0
⊂
In general, not all invariant generalized complex structures on real semisimple Lie groups are regular.
Let G be a finite-dimensional connected real Lie group, k 0 an integer.
0
≥
Definition 2.5. An invariant generalized complex structure on G is called subregular in codimension k
0
J
if the associated subalgebra s g is normalized by a codimension k subalgebra of a Cartan subalgebra of g .
0
⊂
If k 1, then is called subregular strictly in codimension k if is subregular in codimension k, but is
≥ J J
not subregular in codimension k 1.
−
We illustrate this notion with an example of a non-regular invariant generalized complex structure on
SO (2n 1,1), n 3 even.
0
− ≥
3 A non-regular invariant generalized complex structure on SO (2n
0
−
1,1)
Let G = SO (2n 1,1), n 3. Then g = so(2n 1,1) is a noncompact real form of g = so (C). We
0 0 0 2n
− ≥ −
interpret g as the Lie algebra of 2n 2n skew symmetric complex matrices. Then
×
τ: g g, A J A¯ J, J =diag(11 1( 1)),
→ 7→ · · ··· −
2n−1
| {z }
is the conjugation with respect to g , where bar denotes the usual complex conjugation.
0
2
Let E , 1 i,j 2n, be a 2n 2n matrix with 1 in the (i,j)th place and 0 elsewhere. Following [6],
ij
≤ ≤ ×
define
H =√ 1 (E E ), 1 k n,
k 2k−1,2k 2k,2k−1
− · − ≤ ≤
G+ =E E +E E +√ 1 (E E E +E ),
jk 2j−1,2k−1− 2k−1,2j−1 2j,2k− 2k,2j − · 2j−1,2k− 2j,2k−1− 2k,2j−1 2k−1,2j
G+ = G+, 1 j <k n,
kj − jk ≤ ≤
G− =E E E +E +√ 1 (E +E E E ),
jk 2j−1,2k−1− 2k−1,2j−1− 2j,2k 2k,2j − · 2j−1,2k 2j,2k−1− 2k,2j−1− 2k−1,2j
G− = G−, 1 j <k n.
kj − jk ≤ ≤
n
Then h= CH is a Cartan subalgebra of g. Let ǫ h∗, 1 k n, be such that ǫ (H )=1 if j =k
k k k j
L ∈ ≤ ≤
k=1
and 0 otherwise. Then
Φ= ǫ ǫ 1 j =k n (ǫ +ǫ ) 1 j <k n
j k j k
{ − | ≤ 6 ≤ }∪{± | ≤ ≤ }
is the root system of (g,h). Let us choose the following root vectors:
X =X =G+, 1 j =k n,
jk ǫj−ǫk jk ≤ 6 ≤
Y =X =G−, 1 j <k n,
jk ǫj+ǫk kj ≤ ≤
Z =X =G−, 1 j <k n.
jk −(ǫj+ǫk) jk ≤ ≤
Note that
[Y ,Z ]=4 (H +H ), [X ,X ]=4 (H H ), 1 j <k n.
jk jk j k jk kj j k
· · − ≤ ≤
Let h h be a hyperplane cut out by the equation ǫ ǫ =0, L(h a vector subspace containing
1 n−1 n 1
⊂ −
H +H , and H h L. Define
n−1 n 1
∈ \
s=L C(H +X ) CX CY CZ
⊕ n−1,n ⊕ M jk ⊕ M jk ⊕ n−1,n
1≤j<k≤n 1≤j<k≤n
(j,k)6=(n−1,n)
g=h CX (CY CZ ).
⊂ ⊕ M jk ⊕ M jk⊕ jk
1≤j6=k≤n 1≤j<k≤n
Lemma 3.1. The subalgebra s g is subregular strictly in codimension 1.
⊂
Proof. By construction, s is normalized by a codimension 1 subalgebra h h. At the same time, s is not
1
⊂
regular, because, for a suitable l L, l+H +X lies in the radical of s but its nilpotent component
n−1,n
∈
X does not.
n−1,n
Note that s+τ(s)=g if and only if L+CH +τ(L+CH)=h.
n−2
To illustrate the generalidea, let us assume for simplicity that H =H , L= CH C(H +H ).
1 k n−1 n
L ⊕
k=2
Then
n−2
s g = √ 1 b (H +X Z )+ √ 1 b H b R
∩ 0 { − · 1· 1 n−1,n− n−1,n X − · j · j | j ∈ }
j=2
isarealabelianLiealgebraofdimensionn 2. Ifniseven,s g carriesasymplecticformω ,whichmay
0 0
− ∩
n−2
beanynon-degenerate2-formontherealvectorspaces∩g0 =R√−1(H1+Xn−1,n−Zn−1,n)⊕L R√−1Hj ∼=
j=2
Rn−2. One can extend √ 1ω to a closed 2-form ω 2s∗. Assume that ǫ , 1 j n, vanish on the root
0 j
− ∈V ≤ ≤
vectors of g. This proves
3
n−2
Theorem 3.2. Let n 4 be even, s g the complex subalgebra defined above, H = H , L = CH
1 k
≥ ⊂ L ⊕
k=2
C(H +H ), and ω 2s∗ a closed 2-form such that
n−1 n
∈V
n−1
2
ω =√ 1 ǫ ǫ .
|s∩g0 − · X 2j−1∧ 2j
j=1
Then (s,ω) is a g -admissible pair and defines a non-regular invariant generalized complex structure on
0
SO (2n 1,1).
0
−
4 Invariant generalized complex structures on real forms of G
2
In this section, G denotes a connected real Lie group whose Lie algebra g is a real form of the complex
0 0
simple Lie algebra g of type G , i.e. g is either the compact real form Gc or the normal real form Gn of
2 0 2 2
g=G . Let τ: g g be the conjugation with respect to g .
2 0
→
Recall that Gn has 4 conjugacy classes of Cartan subalgebras l: the maximally noncompact, the max-
2
imally compact, the one with a single short real root and the one with a single long real root [13]. The
conjugation τn: g → g with respect to Gn2 acts on the root system of (g,l⊗R C) as Id, −Id, a reflection
through a short root and a reflection through a long root respectively.
Let (s,ω) be a g -admissible pair corresponding to an invariant generalized complex structure on G .
0 0
ThesubalgebrasofthecomplexsimpleLiealgebraoftypeG wereclassifiedin[10]. Wewillusethenotation
2
of [4] and [10]. Since s+τ(s)=g, dim(s) dim(G )/2=7 and s g is regular.
2
≥ ⊂
Lemma 4.1. The subalgebra s g is normalized by a Cartan subalgebra of g and is not isomorphic to
0
⊂
sl (C).
3
Proof. By [4], up to conjugacy either s=g or s=A or s G [β] or s G [α].
2 2 2
⊂ ⊂
Suppose s = A . Since s is semisimple, the 2-dimensional subalgebra s τ(s) contains semisimple and
2
∩
nilpotentcomponentsofits elements. Since H2(s,C)=0, ω is exact. Thus,ifs τ(s) is abelian,ω =0,
∩ |s∩g0
a contradiction. Hence s τ(s) is not abelian, and so every element of s τ(s) is either semisimple or
∩ ∩
nilpotent. Then we can choose a basis x ,x of s τ(s) such that [x ,x ] = 2 x , where x is semisimple
0 1 0 1 1 0
∩ ·
and x is nilpotent. Since τ(x ) Cx , we may assume that τ(x )=x , τ(x )=x .
1 1 1 0 0 1 1
∈
The proof of the Jacobson-Morozovtheorem in [3] goes through and provides x s such that x ,x ,x
2 0 1 2
∈
span an sl (C) subalgebra of s. Since τ(x )=x , we obtain a contradiction.
2 2 2
Suppose s G [β] or s G [α]. By [10], Table 1, s either is solvable and contains a Cartan subalgebra
2 2
⊂ ⊂
of g or is normalized by a Borel subalgebra of g or is the subalgebra
s =h CY CY CY CY CY ,
3 1 β −β 2α+β 3α+β 3α+2β
⊕ ⊕ ⊕ ⊕ ⊕
where h g is a Cartansubalgebra,Φ= α, β, (α+β), (2α+β), (3α+β), (3α+2β) is the root
1
⊂ {± ± ± ± ± ± }
system of (g,h ), Y , γ Φ, are root vectors.
1 γ
∈
Note that any Borel subalgebra b g contains a Cartan subalgebra h0 g0 [15]. Let h=h0 RC.
⊂ ⊂ ⊗
If s b contains a Cartan subalgebra of g, then h is maximally compact and h s = 0. This implies
0
⊂ ∩ 6
that either h normalizes s or h s s τ(s)=0, a contradiction.
∩ ⊂ ∩
Suppose s=s . Let b G [α] be the Borel subalgebra of (g,h ) containing Y and Y , n=[b,b]. Since
3 2 1 α β
⊂
[[n,n],n] s, we may write
⊂
s=h Cx Cx CX CX CX ,
1 3 4 2α+β 3α+β 3α+2β
⊕ ⊕ ⊕ ⊕ ⊕
4
x =a X +a X +X , x =X +b X +b X ,
3 0 α 1 α+β β 4 −β 0 α 1 α+β
· · · ·
where X , γ Φ, are root vectors of (g,h), n contains X and X .
γ α β
∈
We may assume that h =Cx Cx , where
1 1 2
⊕
x =z +X + u X , x =z + v X , z ,z h, ρ(z )=0,
1 1 ρ X γ · γ 2 2 X γ · γ 1 2 ∈ 2
γ≻ρ γ≻ρ
for some ρ α,β,α+β .
∈{ }
Since s [x ,x ]=α(z ) a X +β(z ) X +x , x [n,n], and (α β)(z )=0, a =0. If ρ=α,
2 3 2 0 α 2 β 23 23 2 0
∋ · · · ∈ − 6 6
then also
s (α+β)(z ) a X +β(z ) X ,
2 1 α+β 2 β
∋ · · ·
and so a =0 in this case.
1
If x =X , then [x ,x ]=H +b X . Hence ρ=α, and so [x ,x ] s implies that b =0. Then
3 β 3 4 β 0 α+β 2 4 1
· 6 ∈
s=Cx′ Cx′ Cx CX CX CX CX ,
1⊕ 2⊕ 4⊕ β ⊕ 2α+β ⊕ 3α+β ⊕ 3α+2β
where x′ = z′ +u X , x′ = z′ +v X , z′,z′ h. We may assume that u = 1, v = 0, and so
1 1 · α+β 2 2 · α+β 1 2 ∈
(α+β)(z′)=0.
2
In this case, τ acts on the roots either as Id or as a reflection through β. Hence either z′ or X is
− 2 β
contained in s τ(s)=0, a contradiction.
∩
Hence we may assume that x is not proportionalto a rootvector,andso ρ=α. In this case, s contains
3
x′ =z +X +u X and x′ =z +v X .
1 1 α · α+β 2 2 · α+β
If v =0, we may assume that v =1 and u=0. Since [x′,x′] s, (α+β)(z )=0.
6 1 2 ∈ 1
Since [x′,x ] s, β(z ) X +α(z )b X s, and so b =0.
1 4 ∈ − 1 · −β 1 0· α ∈ 1
Since [x′,x ] s, β(z ) X X s, and so b =1/β(z ).
2 4 ∈ − 2 · −β − α ∈ 0 2
Since [x′,x ] s, β(z ) X X s, and so a = 1/β(z ). Hence
1 3 ∈ 1 · β − α+β ∈ 1 − 1
s=Cx′ Cx′ Cx′ Cx′ CX CX CX ,
1⊕ 2⊕ 3⊕ 4⊕ 2α+β ⊕ 3α+β ⊕ 3α+2β
where x′ =r H r X , x′ =r H r X , x′ =X +r X , x′ =X +r X .
1 1· 3α+β − 2· −β 2 2· 3α+2β − 1· β 3 α+β 1· β 4 α 2· −β
Since τ acts on the roots either as Id or as a reflection throughβ, h CX CX is spanned by x′,
− ⊕ β⊕ −β 1
x′, τ(x′), τ(x′). We can choose the root vectors such that τ(X )= X , γ Φ.
2 1 2 γ ± τ(γ) ∈
Ifτ actsontherootsas Id,thenτ(x′)= r H r X , τ(x′)= r H r X . Hence
− 1 − 1· 3α+β∓ 2· β 2 − 2· 3α+2β∓ 1· −β
CX CX is spanned by a single element (r /r ) X (r /r ) X , a contradiction.
β −β 2 1 −β 2 1 β
⊕ · ± ·
If τ acts on the roots as a reflection through β, then τ(x′) = r H + r X , τ(x′) =
1 ± 1 · 3α+2β 2 · −β 2
r H +r X . Hencehisspannedbyasingleelement(r /r ) H (r /r ) H ,acontradiction.
2 3α+β 1 β 1 2 3α+β 1 2 3α+2β
± · · · ± ·
If v = 0, then u = 0. Since [x′,x ] s, X b X s, and so b = b = 0. Since s τ(s) = 0
2 4 ∈ −β − 1· α+β ∈ 0 1 ∩
and α(z ) = 0, g = Gn and τ = τ acts on the roots as a reflection through β. Hence X s τ(s), a
2 0 2 n −β ∈ ∩
contradiction.
5
Corollary 4.2. The subalgebra s g is normalized by a maximally compact Cartan subalgebra h g .
0 0
⊂ ⊂
Moreover, either s = L n or s = b, where b g is a Borel subalgebra of (g,h), L h = h0 R C and
⊕ ⊂ ⊂ ⊗
n=[b,b].
Proof. Let h0 g0 be a Cartan subalgebra normalizing s, h=h0 RC, Xγ, γ Φ, root vectors of (g,h).
⊂ ⊗ ∈
By Lemma 4.1 and [10], Table 1, the subalgebra s g is one of the following:
⊂
L n, b=h n, CH CX n, CH CX n,
α −α β −β
⊕ ⊕ ⊕ ⊕ ⊕ ⊕
G [β]=CX b, G [α]=CX b, g,
2 −α 2 −β
⊕ ⊕
whereL h,n= CX andΦissuitablyorderedsothatΦ+ = α,β,α+β,2α+β,3α+β,3α+2β Φ
γ
⊂ L { }⊂
γ∈Φ+
is the subset of positive roots.
By [1], Lemma 7, only two of these subalgebras can form a g -admissible pair:
0
s=L n or s=b.
⊕
In both cases, τ acts on the roots as Id.
−
Let φ: G G be the universal complexification [8], i.e. G is the connected complex simple Lie group
0
→
of type G , with Lie algebra g, ker(φ) is the center of G , and the differential of φ is the embedding g g.
2 0 0
⊂
LetB GbeafixedBorelsubgroup,withLiesubalgebrab gcontainingamaximallycompactCartan
⊂ ⊂
subalgebra h g . By [15], Theorem 5.4, H =B G is connected and so is generated by h =b g .
0 0 0 0 0 0
⊂ ∩ ∩
ConsidertheflagmanifoldG/B parametrizingtheBorelsubalgebrasofg. Let ∗ betheholomorphicho-
N
mogeneous vector bundle over G/B corresponding to the isotropy representation B GL(HomC([b,b],C))
→
coming from the adjoint action of B on b.
Let G/B be the union of the open orbits of G . By [15], Theorem 4.5, parametrizes the Borel
0
B ⊂ B
subalgebras containing a maximally compact Cartan subalgebra of g . If G is compact, then = G/B.
0 0
B
Otherwise, consistsof exactlythree openorbits ofG onG/B, correspondingto the three Weylchambers
0
of G contaBined in a Weyl chamber of A +A˜ , [15], Theorem 4.7.
2 1 1
Let
2
= GL(2,R)/GL(1,C), Σ= Σ , where Σ = σ (C2)∗ Im(σ )is symplectic ,
I B× B× 0 0 { ∈^ | |R2 }
be the trivial bundles over parametrizing the complex structures and certain extensions of symplectic
B
structures on the fibers of h respectively, h identified with R2.
0 0
B× →B
Remark 4.3. As sets, GL(2,R)/GL(1,C)= z C Im(z)=0 =Σ .
∼{ ∈ | 6 }∼ 0
Now we state the main result of this section.
Theorem 4.4. Any invariant generalized complex structure on G , a real Lie group of type G and real
0 2
dimension 14, is regular. The set of invariant generalized complex structures on G is parametrized by the
0
disjoint union
,
C∪S
where C =I×BN∗ ∼=N∗|B×GL(2,R)/GL(1,C) and S =Σ×BN∗ ∼=N∗|B×Σ0.
6
Proof. We use the notation of Corollary 4.2.
Suppose s = L n, dim(L) = 1. Then s+τ(s) = g if and only if L h is the holomorphic subspace
⊕ ⊂
ofacomplexstructureonh . Sinces τ(s)=0,anyclosed2-formω 2s∗ givesag -admissiblepair(s,ω).
0 0
∩ ∈V
The Chevalley-Eilenberg resolution gives H2(s,C) = 0. Hence ω = dξ for a uniquely determined linear
map ξ: [b,b] C.
→
Thus, the g -admissible pairs (s,ω) with s=L n are parametrized by the triples (b,ξ,λ), where b g
0
⊕ ⊂
is a Borel subalgebra containing a maximally compact Cartan subalgebra of g0, ξ HomC([b,b],C) and
∈
λ GL(2,R)/GL(1,C) is a complex structure on the real vector space h =b g =R2. Cf. [12].
∈ 0 ∩ 0 ∼
Suppose s=b. In this case, H2(s,C)=C ω , where 0=ω 2h∗ is extended by zero to a 2-form on
0 0
s. Since b g =h , a 2-form ω 2b∗ gives·a g -admissib6 le pa∈irV(b,ω) if and only if ω =c ω +dξ for a
0 0 0 0
∩ ∈V ·
uniquely determined linear map ξ: [b,b] C, where Im(c ω ) is non-degenerate.
→ · 0|h0
Thus, the g -admissible pairs (s,ω) with s=b are parametrized by the triples (b,ξ,σ), where b g is a
0
⊂
Borelsubalgebracontaining a maximally compact Cartansubalgebraofg0, ξ HomC([b,b],C)and σ Σ0.
∈ ∈
As we recalled above, consists of one or three orbits of G [15].
0
B
Corollary 4.5. The set of invariant generalized complex structures on G , up to conjugacy by G , is
0 0
parametrized by r copies of the disjoint union
N∗ GL(2,R)/GL(1,C) N∗ Σ ,
0 × ∪ 0 × 0
where N0∗ =HomC([b,b],C)/H0, r =1 if G0 is compact and 3 otherwise.
The following remark is an immediate consequence of Milburn’s study of invariant generalized complex
structures on homogeneous spaces [11].
Remark 4.6. There is no SO(2n+1)-invariant generalized complex structureon the 2n-dimensional sphere
S2n = SO(2n+1)/SO(2n), n 2, and no G -invariant generalized complex structure on S6 = Gc/SU(3).
≥ 2 2
The SO(3)-invariant generalized complex structures on S2 = SO(3)/SO(2) are two biholomorphic complex
structures CP1 and CP1, and a family of invariant generalized complex structures with holomorphic subbun-
dles of the form L(so (C),ω ), c C, Im(c)=0, which are B-transforms of the symplectic structures (up to
3 c
symplectomorphism) on S2. Nota∈tion is from6 [5], [11], ω 2so (C)∗ is defined by
c 3
∈V
0 c 0
 
ω (X,Y)=Trace( c 0 0 [X,Y])
c
− ·
 0 0 0
for 3 3 skew symmetric complex matrices X,Y so (C). See also [7].
3
× ∈
Acknowledgement
The author is grateful to Beijing International Center for Mathematical Research, the Simons Foundation
and Peking University for support, excellent working conditions and encouraging atmosphere.
References
[1] D. V. Alekseevsky and L. David, Invariant generalized complex structures on Lie groups, Proceedings
of the London Mathematical Society. Third Series 105 (2012), no. 4, 703–729.
7
[2] N. G. Chebotarev, A theorem of the theory of semi-simple Lie groups, Matematicheskii Sbornik N.S.
11(53) (1942), no. 3, 239–244.
[3] D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand
Reinhold Mathematics Series, Van Nostrand Reinhold, New York, 1993.
[4] E.B.Dynkin,Semisimple subalgebras of semisimple Lie algebras, MatematicheskiiSbornikN.S.30(72)
(1952), no. 2, 349–462,(Russian).
[5] M. Gualtieri, Generalized complex geometry, Annals of Mathematics. Second Series 174 (2011), no. 1,
75–123.
[6] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics,
vol. 80, Academic Press, Inc., New York–London, 1978.
[7] N.Hitchin, Generalized Calabi-Yau manifolds, The QuarterlyJournalofMathematics 54(2003),no.3,
281–308.
[8] G. Hochschild, Complexification of real analytic groups, Transactions of the American Mathematical
Society 125 (1966), 406–413.
[9] A.I.Malcev,On the theory of the Lie groups in the large, RecueilMathˆematique(N.S.) 16(58)(1945),
163–190.
[10] E. Mayanskiy, The subalgebras of G , ArXiv preprint, 2016.
2
[11] B. Milburn, Generalized complex and Dirac structures on homogeneous spaces, Differential Geometry
and its Applications 29 (2011), no. 5, 615–641.
[12] H.V.Pittie,The Dolbeault-cohomology ring of a compact, even-dimensional Lie group,IndianAcademy
of Sciences. Proceedings.Mathematical Sciences 98 (1988), no. 2-3, 117–152.
[13] M. Sugiura, Conjugate classes of Cartan subalgebras in real semi-simple Lie algebras, Journal of the
Mathematical Society of Japan 11 (1959), 374–434.
[14] H.-C. Wang, Closed manifolds with homogeneous complex structure, American Journal of Mathematics
76 (1954), 1–32.
[15] J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and
holomorphic arc components, Bulletin of the American Mathematical Society 75 (1969), 1121–1237.
8