402 Submanifolds and Holonomy
dimension of these Cartan subalgebras is called the rank of g. The restriction of the
Killing form B to h ×h is nondegenerate.
Any complex semisimple Lie algebra can be decomposed into the direct sum of
complex simple Lie algebras, which were classified by
´
Elie Cartan. The complex
simple Lie algebras of classical type and rank r are
sl
r+1
(C) , so
2r+1
(C) , sp
r
(C) , so
2r
(C)(r ≥ 3).
The complex simple Lie algebras of exceptional type are
e
6
(C) , e
7
(C) , e
8
(C) , f
4
(C) , g
2
(C) ,
where the index is the rank of the Lie algebra. Note that there are isomorphisms
so
3
(C)=sl
2
(C)=sp
1
(C) , so
5
(C)=sp
2
(C) , so
6
(C)=sl
4
(C).
The Lie algebra so
4
(C)=sl
2
(C) ⊕sl
2
(C) is not simple.
Let h be a Cartan subalgebra of a complex semisimple Lie algebra g. For each
one-form
α
in the dual vector space g
∗
of g we define
g
α
= {X ∈ g :ad(H)X =
α
(H)X for all H ∈ h}.
If
α
= 0andg
α
= {0},then
α
is called a root of g with respect to h and g
α
is called
the root space of g with respect to
α
. The complex dimension of g
α
is always equal
to one. We denote by Δ the set of all roots of g with respect to h. The direct sum
decomposition
g = h ⊕
α
∈Δ
g
α
is called the root spac e decomposition of g with respect to the Cartan subalgebra h.
The set Δ forms a root system of rank r and can be characterized by its Dynkin
diagram. A subset Λ = {
α
1
,...,
α
r
} of Δ is a set of simple roots if every root
α
∈ Δ
can be written as
α
= n
1
α
1
+ ...+ n
r
α
r
with n
1
,...,n
r
∈ Z either all nonnegative or all nonpositive. This induces a disjoint
union Δ = Δ
+
∪Δ
−
. We can then construct a diagram in the following way. To each
simple root
α
i
∈ Λ we assign a vertex which we denote by
. Since the Killing form
B is non-degenerate on h ×h, it induces an inner product on the dual space h
∗
.One
can show that the angle between two simple roots in Λ is one of the following four
angles:
π
2
,
π
3
,
π
4
,
π
6
.
We connect the vertices corresponding to simple roots
α
i
and
α
j
, i = j,by0,1,2or
3 lines if the angle between
α
i
and
α
j
is
π
2
,
π
3
,
π
4
,or
π
6
, respectively. Moreover, if the
vertices corresponding to
α
i
and
α
j
are connected by at least one line and
α
i
,
α
i
>
α
j
,
α
j
, we draw an arrow from the vertex
α
i
to the vertex
α
j
. The resulting object
is called the Dynkin diagram associated with Δ.
We now list the Dynkin diagrams for the complex simple Lie algebras g.Weview
a root system as a subset of an r-dimensional Euclidean vector space V with standard
basis e
1
,...,e
r
.
Basic Material 403
(A
r
) g = sl
r+1
(C), r ≥ 1; V = {v ∈ R
r+1
: v, e
1
+ ...+ e
r+1
= 0};
Δ = {e
i
−e
j
: i = j}; Δ
+
= {e
i
−e
j
: i < j};
α
1
= e
1
−e
2
,...,
α
r
= e
r
−e
r+1
;
α
1
α
2
α
r−1
α
r
(B
r
) g = so
2r+1
(C), r ≥ 2; V = R
r
;
Δ = {±e
i
±e
j
: i < j}∪{±e
i
}; Δ
+
= {e
i
±e
j
: i < j}∪{e
i
};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= e
r
;
α
1
α
2
α
r−2
α
r−1
α
r
+3
(C
r
) g = sp
r
(C), r ≥ 3; V = R
r
;
Δ = {±e
i
±e
j
: i < j}∪{±2e
i
}; Δ
+
= {e
i
±e
j
: i < j}∪{2e
i
};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= 2e
r
;
α
1
α
2
α
r−2
α
r−1
α
r
ks
(D
r
) g = so
2r
(C), r ≥ 4; V = R
r
;
Δ = {±e
i
±e
j
: i < j}; Δ
+
= {e
i
±e
j
: i < j};
α
1
= e
1
−e
2
,...,
α
r−1
= e
r−1
−e
r
,
α
r
= e
r−1
+ e
r
;
α
1
α
2
α
r−3
α
r−2
α
r−1
α
r
o
o
o
o
o
o
O
O
O
O
O
O
(E
6
) g = e
6
(C); V = {v ∈ R
8
: v, e
6
−e
7
= v, e
7
+ e
8
= 0};
Δ = {±e
i
±e
j
: i < j ≤ 5}∪{
1
2
∑
8
i=1
(−1)
n(i)
e
i
∈V :
∑
8
i=1
(−1)
n(i)
even};
Δ
+
= {e
i
±e
j
: i > j}∪{
1
2
(e
8
−e
7
−e
6
+
∑
5
i=1
(−1)
n(i)
e
i
) :
∑
5
i=1
(−1)
n(i)
even};
α
1
=
1
2
(e
1
−e
2
−e
3
−e
4
−e
5
−e
6
−e
7
+ e
8
),
α
2
= e
1
+ e
2
,
α
3
= e
2
−e
1
,...,
α
6
= e
5
−e
4
;
α
1
α
2
α
3
α
4
α
5
α
6
(E
7
) g = e
7
(C); V = {v ∈ R
8
: v, e
7
+ e
8
= 0};
Δ = {±e
i
±e
j
: i < j ≤6}∪{±(e
7
−e
8
)}∪{
1
2
∑
8
i=1
(−1)
n(i)
e
i
∈V :
∑
8
i=1
(−1)
n(i)
even};
Δ
+
= {e
i
±e
j
: i > j }∪{e
8
−e
7
}∪{
1
2
(e
8
−e
7
+
∑
6
i=1
(−1)
n(i)
e
i
) :
∑
6
i=1
(−1)
n(i)
odd};
α
1
=
1
2
(e
1
−e
2
−e
3
−e
4
−e
5
−e
6
−e
7
+ e
8
),
α
2
= e
1
+ e
2
,
α
3
= e
2
−e
1
,...,
α
7
= e
6
−e
5
;
α
1
α
2
α
3
α
4
α
5
α
6
α
7
404 Submanifolds and Holonomy
(E
8
) g = e
8
(C); V = R
8
;
Δ = {±e
i
±e
j
: i < j}∪{
1
2
∑
8
i=1
(−1)
n(i)
e
i
:
∑
8
i=1
(−1)
n(i)
even};
Δ
+
= {e
i
±e
j
: i > j}∪{
1
2
(e
8
+
∑
7
i=1
(−1)
n(i)
e
i
) :
∑
7
i=1
(−1)
n(i)
even};
α
1
=
1
2
(e
1
−e
2
−e
3
−e
4
−e
5
−e
6
−e
7
+ e
8
),
α
2
= e
1
+ e
2
,
α
3
= e
2
−e
1
,...,
α
8
= e
7
−e
6
;
α
1
α
2
α
3
α
4
α
5
α
6
α
7
α
8
(F
4
) g = f
4
(C); V = R
4
;
Δ = {±e
i
±e
j
: i < j}∪{±e
i
}∪{
1
2
(±e
1
±e
2
±e
3
±e
4
};
Δ
+
= {e
i
±e
j
: i < j}∪{e
i
}∪{
1
2
(e
1
±e
2
±e
3
±e
4
};
α
1
= e
2
−e
3
,
α
2
= e
3
−e
4
,
α
3
= e
4
,
α
4
=
1
2
(e
1
−e
2
−e
3
−e
4
);
α
1
α
2
α
3
α
4
+3
(G
2
) g = g
2
(C); V = {v ∈ R
3
: v, e
1
+ e
2
+ e
3
= 0};
Δ = {±(e
i
−e
j
) : i < j}∪{±(2e
i
−e
j
−e
k
) : i = j = k = i};
Δ
+
= {e
1
−e
2
,−2e
1
+ e
2
+ e
3
,−e
1
+ e
3
,−e
2
+ e
3
,−2e
2
+ e
1
+ e
3
,2e
3
−e
1
−e
2
};
α
1
= e
1
−e
2
,
α
2
= −2e
1
+ e
2
+ e
3
;
α
1
α
2
_jt
There is a one-to-one correspondence between the above Dynkin diagrams (or
equivalently, the above root systems) and the complex simple Lie algebras.
Structure theory of compact real Lie groups
Let G be a connected, compact, real Lie group. The Lie algebra g of G admits an
inner product so that each Ad(g), g ∈ G, acts as an orthogonal transformation on g
and each ad(X), X ∈ g, acts as a skewsymmetric transformation on g. This yields the
direct sum decomposition
g = z(g) ⊕[g,g],
where z(g) is the center of g and [g, g] is the commutator ideal in g, which is al-
ways semisimple. The Killing form of g is negative semidefinite. If, in addition, g is
semisimple, or equivalently, if z(g)=0, then its Killing form B is negative definite
and hence −B induces an Ad(G)-invariant Riemannian metric on G.Thismetricis
biinvariant, that is, both left a nd right translations are isometries of G .LetZ(G)
o
be
the identity component of the center Z(G) of G and G
s
be the connected Lie sub-
group of G with Lie algebra [g,g].BothZ(G)
o
and G
s
are closed subgroups of G, G
s
is semisimple with finite center, and G is isomorphic to the direct product Z(G)
o
×G
s
.
A torus in G is a connected abelian Lie subgroup T of G.TheLiealgebrat of
a torus T in G is an abelian subalgebra of g.AtorusT in G that is not properly
contained in any other torus in G is called a maximal torus. Analogously, an abelian
Basic Material 405
subalgebra t of g which is not properly contained in any other abelian subalgebra
of g is called a maximal abelian subalgebra. There is a natural correspondence be-
tween the maximal tori in G and the maximal abelian subalgebras of g. Any maximal
abelian subalgebra t of g is of the form
t = z(g) ⊕t
s
,
where t
s
is a maximal abelian su balgebra of the semisimple Lie algebra [g, g].
Any two maximal abelian subalgebras of g are conjugate via Ad(g) for some g ∈
G. This readily implies that any two max imal tori in G are conjugate. Furthermore,
if T is a maximal torus in G,thenanyg ∈ G is conjugate to some t ∈ T .Anytwo
elements in T are conjugate in G if and only if they are conjugate via the Weyl group
W (G,T ) of G with respect to T .TheWeyl group of G with respect to T is defined by
W (G,T )=N
G
(T )/Z
G
(T ),
where N
G
(T ) is the nor malizer of T in G and Z
G
(T )=T is the centralizer of T in G.
In particular, the conjugacy classes in G are parametrized by T /W(G,T ).Thecom-
mon dimension of the maximal tori of G (resp. of the maximal abelian subalgebras of
g) is called the rank of G (resp. the rank of g). Let t be a maximal abelian subalgebra
of g.Thent(C) is a Cartan subalgebra of g(C). For this reason, t is also called a
Cartan subalgebra of g and the rank of g coincides with the rank of g(C).
We assume from now on that g is semisimple, that is, the center of g is trivial.
Then g is called a compact real form of g(C). Every complex semisimple Lie alge-
bra has a compact real form that is unique up to conjugation by an element in the
connected Lie subgroup of the group of real automorphisms of g(C) with Lie alge-
bra ad(g). The compact real forms of the complex simple Lie algebras are, for the
classical complex Lie algebras,
su
r+1
⊂ sl
r+1
(C) , so
2r+1
⊂ so
2r+1
(C) , sp
r
⊂ sp
r
(C) , so
2r
⊂ so
2r
(C) ,
and, for the exceptional complex Lie algebras,
e
6
⊂ e
6
(C) , e
7
⊂ e
7
(C) , e
8
⊂ e
8
(C) , f
4
⊂ f
4
(C) , g
2
⊂ g
2
(C) .
Let
g(C)=t(C) ⊕
α
∈Δ
(g(C))
α
be the root space decomposition of g(C) with respect to t(C). Each root
α
∈ Δ is
imaginary-valued on t and real-valued o n it. The subalgebra it of t
C
is a real form
of t(C) and we may view each root
α
∈ Δ as a one-form on the dual space (it)
∗
.
Since the Killing form B of g is negative definite, it leads via complexification to a
positive definite inner product on it, which we also denote by B. For each
λ
∈ (it)
∗
there exists a vector H
λ
∈it such that
λ
(H)=B(H,H
λ
)
406 Submanifolds and Holonomy
for all H ∈ it. The inner product o n it induces an inner product ·,· on (it)
∗
.For
each
λ
,
μ
∈ Δ,wethenhave
λ
,
μ
= B(H
λ
,H
μ
).
For each
α
∈ Δ we define the root reflectio n
s
α
: (it)
∗
→ (it)
∗
,
λ
→
λ
−
2
λ
,
α
α
,
α
α
.
The Weyl group of G with respect to T is isomorphic to the group generated by all
s
α
,
α
∈ Δ. Equivalently, we might view W (G,T ) as the group of transformations on
t generated by the reflections in the hyperplanes perpendicular to iH
λ
,
λ
∈ Δ.
Structure theory of real semisimple Lie algebras
Let G be a connected real semisimple Lie group, g its Lie algebra and B its Killing
form. A Cartan involution on g is an involution
θ
on g so that
B
θ
(X,Y )=−B(X,
θ
Y )
is a positive definite inner product on g. Every real semisimple Lie algebra has a
Cartan involution, and any two of them are conjugate by Ad(g) for some g ∈ G.Let
θ
be a Cartan involution on g. Denoting by k the (+1)-eigenspace of
θ
and by p the
(−1)-eigenspace of
θ
,wegettheCartan decomposition
g = k ⊕p.
This decomposition is orthogonal with respect to B and B
θ
, B is negative definite o n
k and positive definite on p,and
[k,k] ⊂ k , [k,p] ⊂ p , [p,p] ⊂ k .
The Lie algebra k ⊕ip is a compact real form of g(C).
Let K be the connected Lie subgroup of G with Lie algebra k. Then there exists
a unique involutive automorphism Θ of G whose differential at the iden tity of G
coincides with
θ
.ThenK is the fixed point set of Θ, is closed, and contains the
center Z(G) of G.IfK is compact, then Z(G) is finite, and if Z(G) is finite, then K is
a maximal compact subgroup of G. Moreover, the map
K ×p → G , (k,X) → kExp(X)
is a diffeomorphism onto G. This is known as a polar decomposition of G.
Let a be a maximal abelian subspace of p.Thenallad(H), H ∈ a,formacom-
muting family of selfadjoint endomorphisms of g with respect to the inner product
B
θ
. For each
λ
∈ a
∗
we define
g
λ
= {X ∈ g :ad(H)X =
λ
(H)X for all H ∈ a}.
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