Download Cohomology and K-theory of Compact Lie Groups

COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
CHI-KWONG FOK
Abstract. In this expository article, we review the computation of the (de Rham) cohomology of compact connected Lie groups and the K-theory of compact, connected and
simply-connected Lie groups.
Contents
1.
Introduction
2
2.
Cohomology of compact Lie groups
3
3.
The cohomology ring structure
11
3.1.
Hopf algebras and their classification
12
3.2.
The map p∗
15
4.
Elements of K-theory
24
5. K-theory of compact Lie groups
25
References
28
Date: July 19, 2010.
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2
CHI-KWONG FOK
1. Introduction
In this expository article we give an account of the computation of the (de Rham)
cohomology and K-theory of compact Lie groups based on the classical work of [CE] and
[A1], as well as the article [R]. These become standard results in the algebraic topology of
compact Lie groups.
For the computation of the cohomology groups of compact Lie groups, we demonstrate
the use of the averaging trick to show that it suffices to compute the cohomology using
left-invariant differential forms, which in turn have a natural correspondence with skewsymmetric multilinear forms on the Lie algebra of the Lie group. In this way the whole
situation is reduced to computing the Lie algebra cohomology. One may further restrict to
the bi-invariant differential forms, the advantage of which is that these forms are automatically closed. We shall then derive some interesting results about the topology of compact
Lie groups using this elegant technique.
As to the cohomology ring structure, we first review the basics of Hopf algebras, which
include the cohomology of compact connected Lie groups as motivating examples. Hopf’s
determination of the general ring structure of cohomology of compact connected Lie groups
by means of a result on the classification of Hopf algebras will then be presented. To obtain
more information about the cohomology ring we shall appeal to the well-known map
(1)
p : G/T × T → G
(g, t) 7→ gtg −1
where T is a maximal torus of G. Using p, we can deduce information about H ∗ (G, R) by
computing H ∗ (G/T, R) and H ∗ (T, R). More precisely, it will be shown that
(2)
p∗ : H ∗ (G, R) → (H ∗ (G/T, R) ⊗ H ∗ (T, R))W
is a ring isomorphism, where W is the Weyl group of G. We will describe the W -module
structure of H ∗ (G/T, R) using Morse theory. Making use of invariant theory, in particular
the famous theorem by Borel that H ∗ (G/T, R) is isomorphic to the space harmonic polynomials on t and Solomon’s result on W -invariants of differential forms on t with polynomial
coefficients(c.f. [So]), Reeder interpreted the right-hand side of (2) as W -invariant subspace
of differential forms with harmonic polynomial coefficients and gave a detailed description
of it(c.f. [R]). It will be shown in the end that we have
Theorem 1.1. If G is compact and connected, then H ∗ (G, R) is an exterior algebra generated by elements from odd cohomology groups.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
3
For K-theory, we will be only concerned about simply-connected compact Lie groups.
The structure of the K-theory is immediate once we know that K ∗ (G) is torsion-free and
apply the fact that rational cohomology ring and rational K-theory of a finite CW -complex
are isomorphic through the Chern character(c.f. [AH]). In fact
Theorem 1.2. If G is a compact, simply-connected Lie group, then K ∗ (G) is an exterior
algebra generated by elements in K −1 (G) induced by fundamental representations of G.
The proof from [A1] will be reproduced, with the technical details coming from K-theory
carefully explained.
2. Cohomology of compact Lie groups
The treatment of this section is mainly based on [CE]. Let G be a compact connected
Lie group. We denote the space of differential n-forms of G by Ωn (G), and use Lg to mean
the map of left multiplication by g. A differential form ω ∈ Ω∗ (G) is left-invariant if
L∗g ω = ω for all g ∈ G
The space of left-invariant n-forms is a subspace of Ωn (G), which will be denoted by ΩnL (G).
Lemma 2.1. If ω ∈ ΩnL (G), then dω ∈ ΩnL (G).
Proof. Note that for all g ∈ G,
L∗g dω = dL∗g ω = dω
m+n
(G).
Lemma 2.2. If ω1 ∈ ΩnL (G), ω2 ∈ Ωm
L (G), then ω1 ∧ ω2 ∈ ΩL
Proof. Simply note that L∗g (ω1 ∧ ω2 ) = L∗g ω1 ∧ L∗g ω2 .
By Lemma 2.1, Ω∗L (G) is a complex with differential d. Let HL∗ (G) be the cohomology
of Ω∗L (G). Lemma 2.2 implies that HL∗ (G) has a ring structure induced by wedge product.
Since Ω∗L (G) is a subspace of Ω∗ (G), there is an inclusion map ι : Ω∗L (G) → Ω∗ (G),
which induces
ι∗ : HL∗ (G) → H ∗ (G, R)
It is plain that ι∗ is a ring homomorphism. In fact we have
Theorem 2.3. ι∗ is a ring isomorphism.
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CHI-KWONG FOK
Before giving a proof of theorem we shall introduce the ‘averaging trick’. Let ω ∈ Ωn (G).
Define
Z
L∗g ωdg
Iω =
G
where dg denotes the normalized Haar measure of G. By the translation invariance of dg,
Iω ∈ ΩnL (G). One can easily show that the averaging operator I is linear, and commutes
with d and pullbacks.
Proof of Theorem 2.3. First we shall show that ι∗ is injective. Suppose that ω ∈ ΩnL (G)
and ω = dµ for some µ ∈ Ωn−1 (G). Then
ω = Iω = Idµ = dIµ
∗
So ω ∈ d(Ωn−1
L (G)) and thus [ω] = 0 in HL (G).
To show that ι∗ is surjective, we shall, for any ω ∈ Ωn (G) with dω = 0, find µ ∈ Ωn−1 (G)
such that ω + dµ ∈ ΩnL (G).
Claim 2.4. There exists µ ∈ Ωn−1 (G) such that ω + dµ = Iω.
Proof. Recall the de Rham theorem, which asserts, for compact manifold M , the nondegenerate pairing between the cohomology class [ω] ∈ H n (M, R) and the homology class
P
[c] ∈ Hn (M, R) represented by a linear combination of n-chains c = i ri Zi , where Zi is
closed n-dimensional submanifolds. The pairing is given by
X Z
ω
h[ω], [c]i =
ri
i
Zi
If we can show that
Z
ω − Iω = 0
(3)
Z
for any n-dimensional submanifold, then [ω − Iω] = 0 and hence ω − Iω is exact.
Note that, since G is connected, there exists a continuous path s : I → G between e and
g. Thus Lg is homotopic to IdG . It follows that the induced map
Lg ∗ : Hn (G, R) → Hn (G, R)
is identity, and that [gZ] = [Z] for any n-dimensional submanifold Z. With this in mind,
Equation (3) can be proved as follows.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
Z Z
Z
Z
ZZ ZG
ZZ
Z
L∗g ωdg
ω−
(ω − Iω) =
ω−
=
Z
G
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L∗g ωdg (Fubini’s theorem)
Z
Z Z
Z
ω−
=
ωdg
G
Z
gZ
Z Z
Z
ω−
=
ωdg
G
Z
Z
=0
Theorem 2.3 implies that it is sufficient to use only the left-invariant forms on G to
compute its cohomology. By way of left invariance, we can further restrict our attention
to skew-symmetric multilinear forms on g, the Lie algebra of G, as one can easily observe
that
Lemma 2.5. The map
ϕ:
Ω∗L (G)
→(
∗
^
g)∗
ω 7→ {ω} := ω|Vn Te G
is a ring isomorphism.
V
Proof. Note that ϕ is linear. Surjectivity is easy: given α ∈ ( n g)∗ , define ω ∈ Ωn (G)
such that
ωg ((X1 )g , · · · , (Xn )g ) = α(Lg−1 ∗ (X1 )g , · · · , Lg−1 ∗ (Xn )g )
for any vector fields Xi , 1 ≤ i ≤ n, on G. ω ∈ ΩnL (G) because
(L∗g ω)h ((X1 )h , · · · , (Xn )h ) = ωgh (Lg ∗ (X1 )h , · · · , (Lg )∗ (Xn )h )
= α(Lh−1 ∗ (X1 )h , · · · , Lh−1 ∗ (Xn )h )
= ωh ((X1 )h , · · · , (Xn )h )
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CHI-KWONG FOK
Let ω ∈ ΩnL (G) such that {ω} = 0. Then for any (X1 )g , · · · , (Xn )g ∈ Tg G,
ωg ((X1 )g , · · · , (Xn )g ) = (L∗g ω)e (Lg−1 ∗ (X1 )g , · · · , Lg−1 ∗ (Xn )g )
= ωe (Lg−1 ∗ (X1 )g , · · · , Lg−1 ∗ (Xn )g )
=0
Note that ϕ is also a ring homomorphism. Together with the above we have shown that
indeed ϕ is a ring isomorphism.
Recall that the differential d on Ωn (G) can be explicitly defined by
(4)
n+1
X
bi , · · · , Xn+1 ) +
(−1)i Xi ω(X1 , · · · , X
dω(X1 , · · · , Xn+1 ) =
i=1
X
i+j
(−1)
bi , · · · , X
bj , · · · , Xn+1 )
ω([Xi , Xj ], X1 , · · · , X
i<j
where X1 , · · · , Xn+1 are vector fields on G.
V
V
Lemma 2.6. The map δ : ( n g)∗ → ( n+1 g)∗ which makes the following diagram commutes
ΩnL (G)
d
ϕ
Vn
(
/ Ωn+1 (G)
L
g)∗
δ
ϕ
V
/ ( n+1 g)∗
is given by
δα(ξ1 , · · · , ξn+1 ) =
X
(−1)i+j α([ξi , ξj ], ξ1 , · · · , ξbi , · · · , ξbj , · · · , ξn+1 )
i<j
Proof. Let (Xi )g = Lg ∗ (ξi ) be the left-invariant vector field which restricts to ξi ∈ Te G ∼
=g
at the identity e. Then
(ϕdω)(ξ1 , · · · , ξn+1 ) = dω(X1 , · · · , Xn+1 )((Xi )g = Lg ∗ (ξi ))
X
bi , · · · , X
bj , · · · , Xn )
=
(−1)i+j ω([Xi , Xj ], X1 , · · · , X
i<j
(The first term of (4) vanishes as both ω and Xi are left invariant)
X
bi )e , · · · , (X
bj )e , · · · , (Xn )e )
=
(−1)i+j {ω}([(Xi )e , (Xj )e ], · · · , (X
i<j
= (δϕω)(ξ1 , · · · , ξn+1 )
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
7
V∗
Theorem 2.7. HL∗ (G) ∼
= H ∗ (( g)∗ , δ)
V
Proof. By Lemma 2.5 and Lemma 2.6, the two complexes (Ω∗L (G), d) and (( ∗ g)∗ , δ) are
isomorphic.
V
Definition 2.8. The cohomology ring H ∗ (( ∗ g)∗ , δ) is called the Lie algebra cohomology
of g, denoted by H ∗ (g).
Corollary 2.9. H ∗ (G, R) ∼
= HL∗ (G) ∼
= H ∗ (g), if G is a compact connected Lie group.
Remark 2.10. Corollary 2.9 says that the cohomology of a compact connected Lie group
is completely determined by local information(i.e. the Lie algebra cohomology of g). This
is made possible by virtue of the symmetry of G, as encapsulated in the trick of averaging
left multiplication and the left invariance of forms in Ω∗L (G).
Corollary 2.11. If g1 and g2 are isomorphic Lie algebras, then H ∗ (G1 , R) ∼
= H ∗ (G2 , R)
where Gi is a compact connected Lie group such that Te Gi = gi , i = 1, 2.
Example 2.12. Since su(2) ∼
= so(3), H ∗ (SU (2), R) ∼
= H ∗ (SO(3), R). Indeed, both are
isomorphic to R[x]/(x2 ).
Remark 2.13. The corollary becomes false if H ∗ (Gi , R) is replaced by H ∗ (Gi , Z), i = 1, 2.
A counterexample is provided by SU (2) and SO(3)(H 2 (SU (2), Z) = 0 while H 2 (SO(3), Z) ∼
=
Z2 ).
Example 2.14. Let G = (S 1 )n , the n-dimensional torus. Then g ∼
= Rn with trivial Lie
bracket. So δ ≡ 0 and
∗
^
H ∗ ((S 1 )n , R) ∼
Rn
=
One can take a step further by considering the complex of bi-invariant differential forms,
i.e. forms that are both left-invariant and right-invariant. Let
Ω∗B (G) = Ω∗L (G) ∩ Ω∗R (G)
denote the space of bi-invariant forms of G. Using the averaging trick as in the proof of
Theorem 2.3, we obtain
Proposition 2.15. The inclusion ι : Ω∗B (G) ,→ Ω∗ (G) induces a ring isomorphism ι∗ :
∗ (G) → H ∗ (G, R).
HB
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CHI-KWONG FOK
V ∗G
Definition 2.16. Let ( ∗ g) denote the subspace invariant under the adjoint action.
Let
∗
^
ψ : Ω∗B (G) → ( g)∗G
map ω to its restriction ωe at the identity.
∗
∗
n
^
^
X
∗G
∗
Proposition 2.17. ( g) = {α ∈ ( g) |
α(ξ1 , · · · , [ξ, ξi ], · · · , ξn ) = 0 for any ξ, ξi ∈
i=1
g, 1 ≤ i ≤ n}
P
Proof. Suppose ni=1 α(ξ1 , · · · , [ξ, ξi ], · · · , ξn ) = 0 for any ξ, ξi ∈ g for 1 ≤ i ≤ n. For
any given g ∈ G, there exists ξ ∈ g such that exp(tξ), 0 ≤ t ≤ 1, joins e and g, as G is
connected. So
d α(Adexp(tξ) ξ1 , · · · , Adexp(tξ) ξn )
dt t=0
=
n
X
α(ξ1 , · · · , [ξ, ξi ], · · · , ξn ) = 0
i=1
It follows that
α(Adg ξ1 , · · · , Adg ξn ) = α(ξ1 , · · · , ξn )
and so α ∈ (
V∗
g)∗G . The converse is easy.
Proposition 2.18. ψ is a ring isomorphism.
Proof. Since ψ is a ring homomorphism, it remains to show that it is both injective and
surjective. Injectivity can be shown in the same way as in the proof of Lemma 2.5. Given
V
α ∈ ( ∗ g)∗G , let ω ∈ Ω∗ (G) be such that
ωg ((X1 )g , · · · (Xn )g ) = α(Lg−1 ∗ (X1 )g , · · · , Lg−1 ∗ (Xn )g )
Obviously ω ∈ ΩL (G). Moreover,
ωgh (Rh∗ (X1 )g , · · · , Rh∗ (Xn )g )
=α(Lh−1 g−1 ∗ Rh∗ (X1 )g , · · · , Lh−1 g−1 ∗ Rh∗ (Xn )g )
=α(Adh−1 Lg−1 ∗ (X1 )g , · · · , Adh−1 Lg−1 ∗ (Xn )g )
=α(Lg−1 ∗ (X1 )g , · · · , Lg−1 ∗ (Xn )g )
=ωg ((X1 )g , · · · , (Xn )g )
Hence ω ∈ ΩR (G).
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
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Similar to Lemma 2.6, we have
Proposition 2.19. The following diagram
ΩnB (G)
d
ψ
Vn ∗G
(
g)
/ Ωn+1 (G)
B
δ
ψ
V
/ ( n+1 g)∗G
commutes
V
Corollary 2.20. The two complexes (Ω∗B (G), d) and (( ∗ g)∗G , δ) are isomorphic, and
V
∗ (G) and H ∗ (( ∗ g)∗G , δ) are isomorphic as well.
hence their cohomology rings HB
V
Proposition 2.21. For any α ∈ ( ∗ g)∗G , δα = 0.
Proof. Fix k. Then
X
cj , · · · , X
ck , · · · , Xn+1 )
(−1)j+k α([Xj , Xk ], X1 , · · · , X
j<k
+
X
ck , · · · , X
cj , · · · , Xn+1 )
(−1)j+k α([Xk , Xj ], · · · , X
j>k
X
ck , · · · , Xn+1 )
=
(−1)k+1 α(X1 , · · · , Xj−1 , [Xk , Xj ], Xj+1 , · · · , X
j<k
+
X
ck , · · · , [Xk , Xj ], · · · , Xn+1 )
(−1)k+1 α(X1 , · · · , X
j>k
=0
by Proposition 2.17
It follows that
0=
n+1
X

X
cj , · · · , X
ck , · · · , Xn+1 )
 (−1)j+k α([Xj , Xk ], X1 , · · · , X
k=1
j<k

+
X
ck , · · · , X
cj , · · · , Xn+1 )
(−1)j+k α([Xk , Xj ], · · · , X
j>k
=2
X
cj , · · · , X
ck , · · · , Xn+1 )
(−1)j+k α([Xj , Xk ], X1 , · · · , X
j<k
= 2(δα)(X1 , · · · , Xn+1 )
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CHI-KWONG FOK
V
V∗
V∗
Corollary 2.22. H ∗ (( ∗ g)∗G , δ) ∼
= ( g)∗G . Hence H ∗ (G, R) ∼
= ( g)∗G .
Actually, Corollary 2.22 does not come in handy for explicit computation of H ∗ (G, R)
in general. We will give a fuller description of H ∗ (G, R) in the next Section. Nonetheless,
we can still deduce lower dimensional cohomology groups of G easily from Corollary 2.22.
Theorem 2.23. Let G be a compact connected semi-simple Lie group. Then H 1 (G, R) =
H 2 (G, R) = 0, H 3 (G, R) 6= 0.
Proof. If α ∈ g∗G , then α([X1 , X2 ]) = 0 for all X1 , X2 ∈ g. Since g is semi-simple, [g, g] = g.
Hence α must be 0 and g∗G = H 1 (G, R) = 0.
V
Let β ∈ ( 2 g)∗G . Then
β([X1 , X2 ], X3 ) + β(X2 , [X1 , X3 ]) = 0
On the other hand, Proposition 2.21 implies that
(δβ)(X1 , X2 , X3 ) = −β([X1 , X2 ], X3 ) + β([X1 , X3 ], X2 ) − β([X2 , X3 ], X1 ) = 0
The above two equations implies that β([X2 , X3 ], X1 ) = 0 for all Xi ∈ g, 1 ≤ i ≤ 3. So
V
( 2 g)∗G = H 2 (G, R) = 0.
Finally, consider the 3-form
γ(X1 , X2 , X3 ) = B([X1 , X2 ], X3 ) ∈ (
3
^
g)∗G
where B is the Killing form of g. Since B is non-degenerate when g is semi-simple, γ is
not zero.
Theorem 2.24. A compact connected Lie group G is semi-simple iff π1 (G) is finite.
Proof. We shall first claim that H 1 (G, R) = 0 iff π1 (G) is finite. Note that H 1 (G, R) = 0
iff H 1 (G, Z) is a torsion abelian group. Since G is compact, H 1 (G, Z) must be finitely
generated. Thus H 1 (G, Z) is a finite abelian group iff H 1 (G, R) = 0. By the universal
coefficient theorem,
H 1 (G, Z) ∼
= Hom(H1 (G, Z), Z) ⊕ Ext(H0 (G, Z), Z) ∼
= Hom(H1 (G, Z), Z)
So H 1 (G, Z) is a finite abelian group iff H1 (G, Z) is also a finite abelian group. But
H1 (G, Z) ∼
= π1 (G)ab = π1 (G)(the last equality holds because π1 (G) is abelian(c.f. [BtD],
Ch. V, Thm. 7.1)). This establishes the claim. If G is compact, connected and semisimple, then by Theorem 2.23, H 1 (G, R) = 0. So π1 (G) is finite by the claim. If π1 (G) is
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
11
finite, then H 1 (G, R) = 0, which in turn implies that [g, g] = g. Note that g is a reductive
Lie algebra and can be written as
g = z(g) ⊕ h
where h is semi-simple. The condition [g, g] = g forces z(g) = 0. This completes the
proof.
Remark 2.25. From the proof of Theorem 2.24, we can see that if G is compact, connected
and semi-simple, then H 1 (G, Z) = 0. To compute H 2 (G, Z), we may again make use of
the universal coefficient theorem to get
H 2 (G, Z) ∼
= Hom(H2 (G, Z) ⊕ Ext(H1 (G, Z), Z)
Since H 2 (G, Z) must be a finite abelian group, and Hom(·, Z) is either a free abelian group
or 0, Hom(H2 (G, Z), Z) = 0 and H 2 (G, Z) ∼
= Ext(H1 (G, Z), Z). But H1 (G, Z) ∼
= π1 (G), so
2
H (G, Z) = π1 (G).
We are now in a position to answer the classical problem that asks which spheres afford
a Lie group structure. It turns out that
Theorem 2.26. S 0 , S 1 and S 3 are the only spheres that can be given Lie group structures.
Proof. We know that S 0 ∼
= Z2 and S 1 ∼
= U (1). If n ≥ 2 and S n affords a Lie group structure,
then it must be semi-simple because π1 (S n ) is trivial. By Theorem 2.23, H 3 (S 3 , R) 6= 0,
so n must be 3. As it turns out, S 3 ∼
= SU (2).
3. The cohomology ring structure
Determining the ring structure of H ∗ (G, R) is a more subtle issue. Nevertheless, a general
description of the algebraic structure of H ∗ (G, R) has been known for a long time. In this
section, we will first review the basics of Hopf algebras, of which the cohomology ring of any
H-space is a motivating example, and a result, due to Hopf, on the classification of Hopf
algebras over a field of characteristic 0, from which one can easily deduce that H ∗ (G, R) is
an exterior algebra on l generators, where l is the rank of G. Next we implement a careful
study of the map (1) and H ∗ (G/T, R) to obtain more information about the l generators,
e.g. the degrees of the generators.
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CHI-KWONG FOK
3.1. Hopf algebras and their classification. Most of the material in this section is
based on the part of Chapter 3 of [H1] on Hopf algebras. Let X be a connected H-space
equipped with a multiplication map µ : X × X → X. For convenience of exposition,
suppose further that R is a commutative ring with identity and that H ∗ (X, R) is a free
R-module. By Künneth formula, H ∗ (X × X, R) is isomorphic to H ∗ (X, R) ⊗R H ∗ (X, R)
as R-algebras. Composing the isomorphism with
µ∗ : H ∗ (X, R) → H ∗ (X × X, R)
we have a graded R-algebra homomorphism
∆ : H ∗ (X, R) → H ∗ (X, R) ⊗R H ∗ (X, R)
It is not hard to show that, if α ∈ H n (X, R), then
∆(α) = α ⊗ 1 + 1 ⊗ α ⊕
k
X
αi0 ⊗ αi00
i=1
where the degrees of
αi0
and
αi00
are less than n.
The above properties of H ∗ (X, R) led Hopf to define an algebra named after him.
Definition 3.1. A Hopf algebra A over R is a graded commutative R-algebra which
satisfies
(1) A0 = R.
(2) There exists a graded algebra homomorphism ∆, called comultiplication,
∆ : A → A ⊗R A
such that if α ∈ An , then
∆(α) = α ⊗ 1 + 1 ⊗ α +
k
X
αi0 ⊗ αi00
i=1
where the degrees of both
αi0
and
αi00
are less than n.
If ∆(α) = α ⊗ 1 + 1 ⊗ α, then α is called a primitive element.
Example 3.2. Let A = R[α], the polynomial algebra over R generated by an element of
even degree. We shall show that it is a Hopf algebra over R. Let us assume that there
exists a comultiplication ∆. Then ∆(α) = α ⊗ 1 ⊕ 1 ⊗ α since there is no element in R[α]
with positive degree less than that of α. It follows that
!
n
X
n
n
n
∆(α ) = (α ⊗ 1 + 1 ⊗ α) =
αi ⊗ αn−i
i
i=1
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
13
∆ thus defined is indeed a comultiplication.
V
Example 3.3. Let A = R (β), the exterior algebra over R generated by one odd degree
element α. Assume that there exists a comultiplication map ∆. Note that
0 = ∆(β 2 )
= ∆(β)2
= (β ⊗ 1 + 1 ⊗ β)2
= β 2 ⊗ 1 + (β ⊗ 1) · (1 ⊗ β) + (1 ⊗ β) · (β ⊗ 1) + 1 ⊗ β 2
=β⊗β−β⊗β
So ∆ indeed defines a comultiplication, and hence A is a Hopf algebra.
Example 3.4. Let A1 and A2 be Hopf algebras over R. Then so is A = A1 ⊗R A2 , with
comultiplication being
∆A = ∆A1 ⊗R ∆A2
As a result, the polynomial algebra R[α1 , · · · , αn ] with degree of αi even for all i, and
V
R (β1 , · · · , βm ) with degree of βj odd for all j, are Hopf algebras over R. So is their
V
tensor product R[α1 , · · · , αn ] ⊗R R (β1 , · · · , βm ).
The following theorem is a partial converse of Example 3.4
Theorem 3.5. Let A be a Hopf algebra over a field F of characteristic 0 such that An is
finite dimensional over F for each n. Then A must be one of the following.
(1) A polynomial algebra F [α1 , · · · , αn ] with degree of αi even for all i.
V
(2) An exterior algebra F (β1 , · · · , βm ) with degree of βj odd for all j.
(3) The tensor product a polynomial algebra as in (1) and an exterior algebra as in (2).
Proof. Since each graded piece An is finite dimensional over F , there exist x1 , · · · , xn , · · ·
such that they generate A and |xi | < |xj | if i < j. Let An be the subalgebra generated by
x1 , · · · , xn . We may assume that xn ∈
/ An−1 . An is a Hopf subalgebra because ∆(xi ) ∈
An ⊗F An for 1 ≤ i ≤ n. Consider the multiplication map
f : An−1 ⊗F F [xn ] → An
^
or f : An−1 ⊗F (xn ) → An
F
if |xn | is even
if |xn | is odd
14
CHI-KWONG FOK
f is surjective by the definition of An . If we can show that f is injective, then An is a
tensor product of a polynomial algebra and an exterior algebra, as An−1 is by inductive
hypothesis.
Let us consider the case where |xn | is even. Suppose that f is not injective, that is,
P
there is a nontrivial relation ki=0 αi xin = 0, with αi ∈ An−1 . We may assume that k is
the minimal degree of the equations of any nontrivial relation between xn and elements of
An−1 . Let I be an ideal in An generated by the positive degree elements in An−1 and x2n ,
and q : An → An /I. Note that xn ∈
/ I. Consider the composition of maps
Id⊗q
∆
An −→ An ⊗F An −→ An ⊗F An /I
We have
(Id ⊗ q) ◦ ∆(αi ) = αi ⊗ 1
(Id ⊗ q) ◦ ∆(xn ) = 1 ⊗ q(xn ) + xn ⊗ 1
=⇒ 0 = (1 ⊗ q) ◦ ∆(
k
X
αi xin )
=
i=0
k
X
(αi ⊗ 1) · (1 ⊗ q(xn ) + xn ⊗ 1)i
i=0
=
k X
i
X
i=0 j=0
=
k
X
i
j
!
αi xjn ⊗ q(xn )i−j
(αi xin ⊗ 1 + iαi xni−1 ⊗ q(xn ))
(q(xin ) = 0 for i ≥ 2)
i=0
=
k
X
!
iαi ⊗
xi−1
n
⊗ q(xn )
i=0
P
Thus we have another relation ki=0 iαi xni−1 = 0 since q(xn ) 6= 0. This relation is nontrivial,
because xn 6= 0, and iαi 6= 0 if αi and i 6= 0 as F is a field of characteristic 0. We get
another nontrivial relation of lower degree, contradicting the minimality of k. Hence the
multiplication map is in fact an algebra isomorphism. The case where |xn | is odd can be
dealt with in a similar way.
Theorem 3.6. If G is a compact connected Lie group of rank l, then H ∗ (G, R) is an
exterior algebra on l generators of odd degrees.
Proof. Note that H ∗ (G, R) satisfies the conditions in Theorem 3.4, and that it is finite
dimensional over R because G is compact. Thus H ∗ (G, R) must be an exterior algebra on
odd degree generators. It remains to show that there are l generators.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
15
Consider the squaring map
f :G→G
g 7→ g 2
Let g0 ∈ G be a regular element, i.e. hg0 i is a maximal torus of G. Then ZG (g0 ) = hg0 i.
Any preimage of g0 under f commutes with g0 and so f −1 (g0 ) ⊂ hg0 i. It follows that
|f −1 (g0 )| = 2l , and deg(f ) = 2l , which implies that f ∗ amounts to multiplication by 2l on
H top (G, R). Suppose H ∗ (G, R) is an exterior algebra on m generators β1 , · · · , βm of odd
degrees. Assume that |βi | < |βj | if i < j. Note that f ∗ = d∗ ◦ ∆, where
d∗ : H ∗ (G × G, R) ∼
= H ∗ (G, R) ⊗ H ∗ (G, R) → H ∗ (G, R)
is induced by the diagonal embedding d : G → G × G and amounts to the wedge product
map. So
f ∗ (βi ) = d∗ ◦ ∆(βi )
= d∗ (1 ⊗ βi + βi ⊗ 1 +
X
0
00
γij
⊗ γij
)
0
00
(|γij
|, |γij
| < |βi |)
j
= 2βi +
X
0 00
γij
γij
j
Note that f ∗ (βi ) = 2βi for i = 1, 2, 3. Since β1 β2 · · · βm ∈ H top (G, R), we have
2l β1 β2 · · · βm = f ∗ (β1 β2 · · · βm )
= (2β1 )(2β2 )(2β3 )(2β4 +
X
0
00
γ4j
γ4j
) · · · (2βm +
j
X
0
00
γmj
γmj
)
j
m
= 2 β1 β2 · · · βm
We conclude that m = l and the proof is complete.
We will give another proof of Theorem 3.6 in Section 3.2.
3.2. The map p∗ . Recall that, if g is a Lie algebra of a compact Lie group G, then gC is
a complex reductive Lie algebra. Let t be the Lie algebra of T , and tC its complexification.
The maps adξ : gC → gC , ξ ∈ gC are simultaneously diagonalizable and give the eigenspace
decomposition of gC
M
gα
gC = t C ⊕
α∈∆
t∗C
where α ∈ ∆ ⊂
are roots of g satisfying [H, Xα ] = α(H)Xα for H ∈ tC , Xα ∈ gα . Note
that dimgα = 1. There exists Hα ∈ tC , α ∈ ∆ and Xα ∈ gα such that
16
CHI-KWONG FOK
(1) α(Hα ) = 2
(2) [Xα , X−α ] = Hα
(c.f. [Se], Ch. VI, Thm. 2), and g is the real span of {iHα , Xα −X−α , i(Xα +X−α )}α∈∆ (c.f.
[H2], Ch. 3, proof of Thm. 6.3). It follows that t = spanR {iHα }α∈∆ , and
M
mα
g=t⊕
α∈∆+
where ∆+ is the set of positive root, and mα = g ∩ (gα ⊕ g−α ) is the real span of the basis
L
{i(Xα + X−α ), Xα − X−α }. We will denote α∈∆+ mα by m. The matrix representation
of the action ad(H) on mα with respect to this basis is
!
0
−iα(H)
iα(H)
0
whereas that of the adjoint action Adexp(H) is
cos iα(H) − sin iα(H)
sin iα(H) cos iα(H)
!
Theorem 3.7. If TgT G/T is identified with m, Tt T with t and Tgtg−1 G with g by left
translation, then
dp(gT,t) : m ⊕ t → g
(X, T ) 7→ Adgt−1 g−1 (X) − X + Adg (T )
In matrix form,
dp(gT,t) =
0
Adgt−1 g−1 − Idm
0
Adg |t
!
Proof. Identify (X, T ) ∈ m ⊕ t with (Lg∗ X, Lt∗ T ) ∈ TgT G/T ⊕ Tt T . Then
d dp(gT,t) (Lg∗ X, Lt∗ T ) =
gexp(sX)texp(sT )g −1 exp(−sX)
ds s=0
= Rtg−1 ∗ Lg∗ X + Rg−1 ∗ Lgt∗ T − Lgtg−1 ∗ X
= Lgtg−1 ∗ (Adgt−1 g−1 X − X + Adg T )
The last line is identified with Adgt−1 g−1 X − X + Adg T .
Corollary 3.8. det dp(gT,t) = det(Adt − Idm ).
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
17
Proof. Note that det(Adg ) = 1 for all g ∈ G because µ : G → R× defined by µ(g) =
det(Adg ) is a group homomorphism and G is compact connected. By Theorem 3.7,
det dp(gT,t) = det(Adgt−1 g−1 − Idm ) det(Adg |t )
= det(Adg (Adt−1 − Idm )Adg−1 )
= det(Adt−1 − Idm )
= det(Adt (Adt−1 − Idm ))
= det(Idm − Adt )
= det(Adt − Idm ) (dimm is even)
Lemma 3.9. If t = exp H, H ∈ t, then det(Adt − Idm ) =
Q
α∈∆+
4 sin2
iα(H)
2 .
Proof.
cos iα(H) − 1 − sin iα(H)
det(Adt − Idm ) =
det
sin iα(H)
cos iα(H) − 1
α∈∆+
Y
=
2(1 − cos iα(H))
!
Y
α∈∆+
=
Y
α∈∆+
4 sin2
iα(H)
2
Suppose g0 ∈ G is a regular value of p. It is well-known that p−1 (g0 ) consists of |W |
points. By Lemma 3.9, the determinant of dp at each point in the pre-image must be
positive. Hence degp = |W |. The pull-back formula for integration gives
Theorem 3.10 (Weyl integration formula). Let ωG , ωG/T and ωT be the normalized volume
form of G, G/T and T respectively, and f : G → C a continuous complex-valued function
on G. Then
Z
Z
1
f (g)ωG =
f ◦ p(gT, t) det(Adt − Idm )ωG/T ∧ ωT
|W | G/T ×T
G
In particular, if f is a class function on G, then
Z
Z
1
f (g)ωG =
f (t) det(Adt − Idm )ωT
|W | T
G
Lemma 3.11. dimR H ∗ (G, R) = 2l
18
CHI-KWONG FOK
V
V
V
Proof. By Corollary 2.22, dimH ∗ (G, R) = dim( ∗ g)∗G = dim( ∗ g)G . Note that ( ∗ g)G
V
is the trivial subrepresentation of the adjoint representation of G on ∗ g. Let χV∗ g be
the character of this representation. Then
Z
∗
^
dim( g)G =
χV∗ g (g)ωG
G
Z
=
χV∗ g (t) det(Adt − Idm )ωT
T
Z
det(Adt + Idm ) det(Adt − Idm )ωT
=
T
Z
=
det(Adt2 − Idm )ωT
T
Z
dimT
=2
det(Ads − Idm )ωs
T
l
=2
Now that G and G/T × T are manifolds of the same dimension, and degp = |W | 6= 0,
the induced map
p∗ : H ∗ (G, R) → H ∗ (G/T × T, R)
is injective. There is a W -action on G/T × T defined by
w · (gT, t) = (gw−1 T, wtw−1 )
and it is easy to see that p(gT, t) = p(w·(gT, t)) for w ∈ W . Thus Im(p∗ ) ⊆ H ∗ (G/T ×T )W .
By abuse of notation we also use p∗ to mean the map
p∗ : H ∗ (G, R) → H ∗ (G/T × T, R)W
We claim that
Theorem 3.12. p∗ is a ring isomorphism.
Before giving a proof of Theorem 3.12, we shall examine H ∗ (G/T, R) more closely. As
a first shot, we shall employ Morse theory to compute the cohomology groups.
Let h·, ·i be an Ad(G)-invariant inner product on g. This can be obtained by averaging
any inner product on g over G. Let X ∈ t+ be a regular element in the positive Weyl
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
chamber, i.e.
α(X)
i
19
> 0 for all α ∈ ∆+ . We let
f : G/T → R
gT 7→ hAdg (X), Xi
and claim that it is a Morse function. Suppose g0 T is a critical point of f . Then for all
Y ∈ m,
0 = df (Lg0 ∗ Y )
d f (g0 exp(sY )T )
=
ds s=0
d =
hAdg0 exp(sY ) (X), Xi
ds s=0
= hAdg0 ([Y, X]), Xi
= h[Y, X], Adg−1 Xi
0
= hY, [X, Adg−1 X]i by Ad-invariance of the inner product
0
Note that
m⊥
= t, as
ht, mi = ht, [t, m]i = h[t, t], mi = 0
Thus [X, Adg−1 (X)] ∈ t, and Adg−1 X ∈ t. g0−1 T and hence g0 T must be a Weyl group
0
0
element. The critical points of f are therefore all the Weyl group elements.
The Hessian Hw (Y, Z) for Y, Z ∈ m, w = g0 T ∈ W is
d hY, [X, Ad(g0 exp(tZ))−1 (X)]i
dt t=0
=hY, [X, [−Z, Adg−1 (X)]]i
0
= − h[Y, X], [Z, Adg−1 (X)]i
0
= − h[X, Y ], [Adg−1 (X), Z]i
0
For α ∈
∆+ ,
Hw (Xα − X−α , Xα − X−α ) = α(X)α(Adg−1 (X))hi(Xα + X−α ), i(Xα + X−α )i =
6 0
0
Hw (i(Xα + X−α ), i(Xα + X−α )) = α(X)α(Adg−1 (X))hXα − X−α , Xα − X−α i =
6 0
0
So Hw is nondegenerate for all w ∈ W and f is indeed a Morse function. The index of f
at w is twice the number of positive roots α such that
α(X)α(Adg−1 (X)) < 0
0
20
CHI-KWONG FOK
Since α(X)
> 0, and α(Adg−1 (X)) = (w · α)(X), the index is also twice the number
i
0
of positive roots α such that w · α is also positive. Let Index(w) = 2m(w). Then the
P
Poincaré polynomial of H ∗ (G/T, R) is P (t) = w∈W t2m(w) , and the Euler characteristic
is χ(G/T ) = P (−1) = |W |.
The W -action on G/T given by w · (gT ) = gw−1 T induces a W -representation on
H ∗ (G/T, R). The trace of w on H ∗ (G/T, R) is just the Lefschetz number of the action
of w because H ∗ (G/T, R) is concentrated on even degrees. If w 6= 1, then it has no fixed
points on G/T , and therefore its trace is 0 by Lefschetz Fixed Points Theorem. If w = 1,
then the trace is just the Euler characteristic |W |. As a result,
Proposition 3.13. H ∗ (G/T, R) is a regular representation of W .
Proof of Theorem 3.12. Since p∗ is injective, it suffices to show that dimH ∗ (G/T ×T, R)W =
dimH ∗ (G, R) = 2l . By Künneth formula, H ∗ (G/T ×T, R)W = (H ∗ (G/T, R)⊗H ∗ (T, R))W .
So
dim(H ∗ (G/T, R ⊗ H ∗ (T, R))W
1 X
=
χH ∗ (G/T,R)⊗H ∗ (T,R) (w)
|W |
w∈W
1 X
=
χH ∗ (G/T,R (w)χH ∗ (T,R) (w)
|W |
w∈W
=
1
χ ∗
(1)χH ∗ (T,R) (1)
|W | H (G/T,R)
=dimH ∗ (T, R)
=2l
Let S be the real polynomial ring S ∗ (t∗ ) on t, and I be the ideal in S generated by
W -invariant polynomials. A famous theorem of Borel describes the ring structure of
H ∗ (G/T, R) using
Theorem 3.14 (Borel). There is a degree-doubling W -equivariant ring isomorphism
c : S/I → H ∗ (G/T, R)
where c(λ)(X, Y ) = λ([X, Y ]) for X, Y ∈ m and deg(λ) = 1.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
21
We refer the reader to [R] for a proof of Theorem 3.14 using invariant theory. Here we
would like to show that H ∗ (G/T, R) is isomorphic to S/I using equivariant cohomology.
For a review of equivariant cohomology the reader is refer to the Appendix.
Consider G/T with T acting on it by left translation. Then
HT∗ (G/T, R) ∼
= HT∗ ×T (G, R)
where T × T acts on G by
(t1 , t2 ) · g = t1 gt2
Next note that G is diffeomorphic to the orbit space of the G-action on G × G given by
g · (g1 , g2 ) = (g1 g, g −1 g2 ). We get
HT∗ ×T (G, R) ∼
= HT ×T ×G (G × G, R)
where T × T × G acts on G × G by
(t1 , t2 , g) · (g1 , g2 ) = (t1 g1 g, g −1 g2 t2 )
So
∗
HT∗ ×T ×G (G × G, R) ∼
(G/T × G/T, R)
= HG
∗
∗
∼
(G/T, R) ⊗HG∗ (pt,R) HG
(G/T, R)
= HG
∼ H ∗ (pt, R) ⊗H ∗ (pt,R) H ∗ (pt, R)
=
T
G
T
It is well-known that
HT∗ (G/T, R) ∼
= HT∗ (pt, R) ⊗R H ∗ (G/T, R)
as HT∗ (pt, R)-modules, because G/T is a T -Hamiltonian manifold. Therefore
H ∗ (G/T, R) ∼
= R ⊗HG∗ (pt,R) HT∗ (pt, R)
∼
= HT∗ (pt, R)/hr∗ HT (pt, R)i
∗ (pt, R) → H ∗ (pt, R) is the map induced by restricting G-action to T -action.
where r∗ : HG
T
By the abelianization principle(c.f. [AB]), r∗ is injective and its image is HT∗ (pt, R)W .
Identifying HT∗ (pt, R) with S, we get Theorem 3.14. Combining Theorem 3.12 and 3.14,
V
and regarding H ∗ (T, R) = ∗ t∗ as differential forms, we have
V∗ ∗ W
Theorem 3.15. H ∗ (G, R) ∼
t ) where the RHS is the space of W = ((S/I)(2) ⊗
invariant differential forms with coefficients in S/I. Here (S/I)(2) means S/I with degree
of each polynomial doubled.
22
CHI-KWONG FOK
It is a classical result in invariant theory, due to Chevalley, that the W -invariant polynomial S W is generated by l algebraically independent polynomials F1 , · · · , Fl . In other
words
S W = R[F1 , · · · , Fl ]
Definition 3.16. The exponents mi , 1 ≤ i ≤ l of G are defined to be
mi = degFi − 1
Q
= 12 dimG/T and li=1 (1 + mi ) = |W |.
V
Theorem 3.18 (Solomon [So]). The space (S ⊗ ∗ t∗ )W of W -invariant differential forms
with polynomial coefficients is an exterior algebra over S W generated by dF1 , · · · , dFl .
Remark 3.17. It is known that
Pl
i=1 mi
Before proving Theorem 3.18, we need a
Lemma 3.19. Let J = Jac(F1 , · · · , Fl ). Then w · J = det(w)J. Here det(w) means the
determinant of w as a linear transformation on t∗ . If R ∈ S and satisfies w · R = det(w)R,
then R = SJ for some S ∈ S W .
Proof. A classical result in invariant theory asserts that, if α1 , · · · , αl ∈ t∗ are simple roots,
J = cα1 α2 · · · αl
for some c ∈ R. Let u ∈ S such that w · u = det(w)u. As {α1 , · · · , αl } forms a basis
for t∗ , those simple roots can be regarded as coordinate functions on t, and we write
u = u(α1 , · · · , αl ), a polynomial of α1 , · · · , α. Note that
u(α1 , · · · , αl−1 , 0) = u(α1 (Hl ), · · · , αl−1 (Hl ), αl (Hl ))
= u(α1 (sHl (Hl )), · · · , αl−1 (sHl (Hl )), αl (sHl (Hl )))
= (sαl · u)(α1 (Hl ), · · · , αl (Hl ))
= −u(α1 , · · · , αl−1 , 0)
So u(α1 , · · · , αl−1 , 0) = 0, and αl divides u. Similar reasoning implies that αi divides u for
1 ≤ i ≤ l. Since α1 , · · · , αl are coprime, J divides u.
Proof of Theorem 3.18. Let L = S(0) be the field of fraction of S. We shall first show that
P
{dFi1 ∧ · · · ∧ dFir }1≤r≤l is linearly independent over S and L. Suppose
ki1 ···ir dFi1 ∧ · · · ∧
dFir = 0. Multiplying dFj1 ∧ · · · ∧ dFjl−r , where {j1 , · · · , jl−r } = {1, · · · , l} \ {i1 , · · · , ir },
we have
±ki1 ···ir Jdx1 ∧ · · · ∧ dxl = 0
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
23
Since F1 , · · · , Fl are algebraically independent, J 6= 0. Thus ki1 ···ir = 0 as both S and L
have no zero divisors.
V
Let ω ∈ (S ⊗ p t∗ )W . As S ⊂ L, we may regard ω as a W -invariant differential p-form
Vp ∗
with coefficients
being
rational
functions
on
t.
Note
that
L
⊗
t is an L-vector space of
!
V
l
dimension
, and thus {dFi1 ∧ · · · ∧ dFip } spans L ⊗ p t∗ over L. Write
p
X si ···i
r
1
ω=
dFi1 ∧ · · · ∧ dFir
ti1 ···ir
s
···ir
where si1 ···ir and ti1 ···ir ∈ S. By the W -invariance of ω, tii1···i
is also W -invariant. Multir
1
plying dFj1 ∧ · · · ∧ dFjl−r with {j1 , · · · , jl−r } = {1, · · · , l} \ {i1 , · · · , ir }, we get
ui1 ···ir dx1 ∧ · · · ∧ dxl =
si1 ···ir
Jdx1 ∧ · · · ∧ dxl
ti1 ···ir
for some ui1 ···ir ∈ S. Comparing coefficients, we have
ui1 ···ir ti1 ···ir = si1 ···ir J
Note that w · ui1 ···ir = det(w)ui1 ···ir . By Lemma 3.19, ui1 ···ir = Pi1 ···ir J for some Pi1 ···ir ∈
s ···ir
and the proof is complete.
S W . It follows that Pi1 ···ir = tii1···i
1
Corollary 3.20. (S/I ⊗
V∗
r
t∗ )W is an exterior algebra over R with generators
dFi ∈ ((S/I)mi ⊗
1
^
t∗ ) W
where dFi means dFi with polynomial coefficients modulo I.
Theorem 3.21 ([R]). If G is compact and connected, then H ∗ (G, R) is an exterior algebra
V
generated by elements of degree 2mi + 1, which correspond to dFi ∈ ((S/I)2mi ⊗R 1 t)W
with the degree of polynomial coefficients doubled.
Example 3.22. Let G = SU (n), and T be the subgroup of diagonal matrices, which is of
dimension n − 1.





ix


1
n


X

.


.
x
∈
R,
x
=
0
t= 
.
i
i





i=1


ixn S∼
= R[x1 , · · · , xn ]/hx1 + · · · + xn i
SW ∼
= R[σ2 , · · · , σn ]
24
CHI-KWONG FOK
P
Q
where σi = 1≤j1 <···<ji ≤n ik=1 xjk is the i-th elementary symmetric polynomial. Hence
the exponents mi are 1, 2, · · · , n − 1. By Theorem 3.14,
H ∗ (G/T, R) ∼
= R[x1 , · · · , xn ]/hσ1 , · · · , σn i ∼
= R[x1 , · · · , xn ]/h(1 + x1 ) · · · (1 + xn ) = 1i
If we think of SU (n)/T as the full flag manifold Fl(Cn ) = {0 = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂
Vn−1 ⊂ Vn = Cn |dimVi = i}, then xi = c1 (Li /Li−1 ), where
Li = {((V0 , V1 , · · · , Vn ), v) ∈ Fl(Cn ) × Cn |v ∈ Vi }
and by Whitney Product Formula, the single relation (1 + x1 ) · · · (1 + xn ) = 1 translates
L
to the fact that the direct sum ni=1 Li /Li−1 is isomorphic to the trivial rank n complex
vector bundle on Fl(Cn ). By Theorem 3.21, H ∗ (SU (n), R) is an exterior algebra on n − 1
generators of degrees 3, 5, · · · , 2n − 1.
4. Elements of K-theory
Let X be a compact topological space and Vect(X) be the category of isomorphism
classes of (finite rank) complex vector bundles over X. Note that Vect(X) is a monoid
with direct sum being binary operation. Let
S(X) = {[E] − [F ]|[E], [F ] ∈ Vect(X)}
We say [E1 ] − [F1 ] ∼ [E2 ] − [F2 ] if there exists [G] ∈ Vect(X) such that [E1 ⊕ F2 ⊕ G] =
[E2 ⊕ F1 ⊕ G].
Definition 4.1. K(X) := S(X)/ ∼. In other words, K(X) is the Grothendieck group of
Vect(X).
Definition 4.2. Let dim : K(X) → Z be the group homomorphism which sends the class
e
of a vector bundle to its rank. Define the reduced K-theory K(X)
to be the kernel of dim.
e q ∧ X), where ∧ means the smash
Definition 4.3. K 0 (X) := K(X), K −q (X) := K(S
product.
L∞
−q
One can make
q=0 K (X) into a ring using tensor product of vector bundles(c.f.
[H1]). The renowned Bott periodicity states that K −q (X) ∼
= K −q−2 (X) for all q ≤ 0.
Definition 4.4. K ∗ (X) := K 0 (X) ⊕ K −1 (X), with ring structure induced by tensor
product of vector bundles.
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
25
K ∗ is a Z2 -graded generalized cohomology theory. Let U (∞) be the direct limit of U (n)
as n tends to infinity, where the morphism U (n) → U (m) for n ≤ m is inclusion. It is
well-known that Z × BU (∞) and U (∞) are classifying spaces of K 0 and K −1 respectively.
Example 4.5. K ∗ (S 2 ) ∼
= Z[H]/(H − 1)2 as rings, where 1 ∈ Z represents the trivial line
bundle and H = O(1).
5. K-theory of compact Lie groups
From now on we assume that G is a simply-connected compact Lie group. This section
is mainly taken from [A1] and [H3]. Let ρ1 , · · · , ρl be the l fundamental representations of
G. A representation ρ : G → U (n), composed with the inclusion U (n) ,→ U (∞), defines
an element in K −1 (G), which we denote by β(ρ). The main result of this section is
Theorem 5.1. If G is a simply-connected compact Lie group, then
^
K ∗ (G) ∼
= (β(ρ1 ), · · · , β(ρl ))
We postpone the proof of Theorem 5.1 to the end of this section. Consider p∗ : K ∗ (G) →
K ∗ (G/T ×T ). Note that K ∗ (G/T ) is torsion-free because G/T can be given a CW-complex
structure consisting of only even-dimensional cells. The same is true of K ∗ (T ) by Lemma
5.8. By Künneth formula for K-theory, K ∗ (G/T × T ) ∼
= K ∗ (G/T ) ⊗ K ∗ (T ).
Lemma 5.2. Consider p∗ : K ∗ (G) → K ∗ (G/T ×T ) ∼
= K ∗ (G/T )⊗K ∗ (T ). Then p∗ (β(ρ)) =
Pn
i=1 α(µj ) ⊗ β(µj ) where µj ’s are all the weights of ρ, and
α : R(T ) → K ∗ (G/T )
is defined by α(µ) = [G ×T Cµ ].
e
Proof. Note that K −1 (G) ∼
where S(G) is the unreduced suspension of G. We
= K(S(G)),
would like to construct β(ρ) explicitly as a (virtual) vector bundle over S(G). Consider
the principal G-bundle G ∗ G → S(G), where G ∗ G = {sg1 + (1 − s)g2 |g1 , g2 ∈ G, s ∈ [0, 1]}
and G acts on G ∗ G by g · (sg1 + (1 − s)g2 ) = sg1 g + (1 − s)g −1 g2 . Note that it is the
pullback of EG → BG through the canonical embedding
S(G) = G ∗ G/G ,→ BG = lim G
∗ ·{z
· · ∗ G} /G
n→∞ |
n times
. We have that
β(ρ) = [(G ∗ G) ×G Vρ ] − [Cdim(ρ) ]
26
CHI-KWONG FOK
Consider the following diagram
G × (T ∗ T )
θ
G/T × S(T )
h
m
/ G∗G
ψ
f
/ G ∗ G/T
q
S(G/T × T )
Sp
/ S(G)
where m is defined by (g, st1 + (1 − s)t2 ) 7→ sgt1 + (1 − s)t2 g −1 , the various vertical maps
projection maps of fiber bundles, and f and Sp are defined in such a way that the above
diagram commutes. Note that
f ∗ [(G ∗ G) ×T Cµ ] = [(G × (T ∗ T ) ×T ×T Cµ ] = α(µ) ⊗ (1 + β(µ))
h∗ (Sp)∗ [(G ∗ G) ×G Vρ ] = f ∗ q ∗ [(G ∗ G) ×G Vρ ]
n
X
= f∗
!
[(G ∗ G) ×T Cµi ]
i=1
=
n
X
α(µi ) ⊗ (1 + β(µi ))
i=1
= [C
dim(ρ)
]+
n
X
α(µi ) ⊗ β(µi )
i=1
Q
Definition 5.3. Let a := li=1 β(ρi ) ∈ K ∗ (G).
Z
Proposition 5.4.
ch(p∗ (a)) = |W |.
G/T ×T
P
Proof. By Lemma 5.2, p∗ (β(ρi )) = m
j=1 α(λij ) ⊗ β(λij ), where λij are the weights of the
fundamental representation ρi and mi is its dimension. We first prove a
Claim 5.5.
mi
l X
Y
i=1 j=1
where ρ =
1
2
P
α∈R+
!
(λij ⊗ β(λij )) =
X
det(w)w · ρ
⊗
w∈W
α and $i is the i-th fundamental weight.
l
Y
i=1
β($i )
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
We have
Z

ch(p∗ (a)) =
G/T ×T
Z
G/T ×T
ch 
mi
l X
Y

α(λij ) ⊗ β(λij )
i=1 j=1


Z
=
G/T ×T
ch (α ⊗ Id) 
mi
l X
Y

(λij ⊗ β(λij ))
i=1 j=1
Z
X
ch (α ⊗ Id)
=
G/T ×T
det(w)w · ρ ⊗
!!
X
ch α
G/T ×T
det(w)w · ρ
⊗ ch
!!
X
ch α
=
G/T
!!
β($i )
l
Y
!
β($i )
i=1
w∈W
Z
l
Y
i=1
w∈W
Z
=
27
det(w)w · ρ
Z
×
ch
T
w∈W
l
Y
!
β($i )
i=1
= χ(G/T ) · 1
= |W |
Z
ch(a) = 1.
Proposition 5.6.
G
Proof.
Z
deg(p)
Z
ch(a) =
ch(p∗ (a)) = |W |
G/T ×T
G
and deg(p) = |W |.
Lemma 5.7 (a special case of [H3], Theorem A(i)). K ∗ (G) is torsion-free.
Before proving Lemma 5.7 we show
Lemma 5.8 ([H3]). If X is a finite CW -complex, and K ∗ (X) has p-torsion, then so does
H ∗ (X, Z).
Proof. Let Qp be Z localized at the prime p. If H ∗ (X, Z) has no p-torsion, then the
homomorphism of spectral sequences for K ∗ (X) ⊗ Qp and K ∗ (X) ⊗ Q
Er (X, Q) → Er (X, Q)
is injective when r = 2, as E2 (X, Qp ) = H ∗ (X, Qp ) and E2 (X, Q) = H ∗ (X, Q). By a result
of Atiyah-Hirzebruch’s(c.f. [AH], p. 19), Er (X, Q) collapses on the E2 -page. Induction on
28
CHI-KWONG FOK
r gives that Er (X, Qp ) also collapses on the E2 -page. Now the associated graded group of
K ∗ (X) ⊗ Qp is E2 (X, Qp ) = H ∗ (X, Qp ) which has no p-torsion. Therefore K ∗ (X, Qp ) has
no p-torsion and so does K ∗ (X).
Sketch of proof of Lemma 5.7. By virtue of Lemma 5.8, it suffices to show that, even if
H ∗ (G, Z) has p-torsion, then K ∗ (G) has no p-torsion. This can be done using a result of
Borel’s which give an exhaustive list of prime p and simple, simply-connected compact Lie
group G such that H ∗ (G, Z) has p-torsion, and showing that K ∗ (G) has no p-torsion case
by case. We refer the reader to [H3], III.1 for detailed proof.
Proof of Theorem 5.1. From the proof of Theorem 3.12, we know dimH ∗ (G, Q) = 2l .
Chern character gives a ring isomorphism between K ∗ (G) ⊗ Q and H ∗ (G, Q)(c.f. [AH]).
As K ∗ (G) is torsion-free by Lemma 5.7, it is a free abelian group of rank 2l . Let Λ =
V
(e1 , · · · , el ) be the exterior algebra generated by e1 , · · · , el over Z. Define
j : Λ → K ∗ (G)
ei 7→ β(ρi )
Proposition 5.6 implies that j is an injective ring homomorphism. Since both Λ and K ∗ (G)
have the same rank, j(Λ) has a finite index in K ∗ (G). Note that one can use the K-theory
pushforward(or the index map)
f! : K ∗ (G) → K ∗ (pt) ∼
=Z
R
R
defined by f! ([E]) = G ch(E)td(G) = G ch(E)(td(G) = 1 as G is parallelizable), to define
k : K ∗ (G) → Hom(Λ, Z)
such that
k(x)(y) = f! (xj(y)), x ∈ K ∗ (G), y ∈ Λ
V
V
We have k ◦ j( i∈I ei )( j∈J ej ) = ±1 if I ∪ J = {1, · · · , l}. So k ◦ j is an isomorphism. It
follows that j(Λ) is a direct summand of K ∗ (G). Being of finite index in K ∗ (G), j(Λ) is
actually isomorphic to K ∗ (G). This completes the proof.
References
[A1]
[A2]
[AB]
M. F. Atiyah, On the K-theory of compact Lie groups, Topology, Vol. 4, 95-99, Pergamon Press,
1965.
M. F. Atiyah, K-theory, W. A. Benjamin, Inc., 1964.
M. F. Atiyah, R. Bott,
COHOMOLOGY AND K-THEORY OF COMPACT LIE GROUPS
[AH]
29
M. F. Atiyah, F. Hirzebruch, Vector bundles and homogeneous spaces, Proceedings of Symposia in
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[BtD] T. Bröcker, T. tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics,
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[CE] C. Chevalley, S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Transactions of the
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[GKM]
[GS] V. Guillemin and S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Springer-Verlag
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http://www.math.cornell.edu/ hatcher/VBKT/VBpage.html
[H2] S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, Vol. 34, AMS, 2001
[H3] L. Hodgkin, On the K-theory of Lie groups, Topology Vol.6, pp. 1-36, Pergamon Press, 1967.
[R]
M. Reeder, On the cohomology of compact Lie groups, L’Enseignement Mathématique, Vol. 41, 1995.
[Se]
J-P. Serre, Complex semisimple Lie algebras, translated from the French by G. A. Jones, Springer
Monograph in Mathematics, Springer-Verlag Berlin Heidelberg, 2001
[So] L. Solomon, Invariants of finite reflection groups, Nagoya J. Math. 22, 57-64, 1963.