Introduction to representation theory
of compact Lie groups
Reimundo Heluani
March 2013, Curitiba
A Group is a tuple (G, e, ·, ·−1), where
• G is a set
• e : {∗} → G is an element of G.
• · : G × G → G is function called multiplication
• ·−1 : G → G is a function called inversion.
Subject to the following axioms
• Associativity
G×G×G
Id×·
/
G×G
·×Id G×G
/
·
·
G
• Left (right) Unit
e×Id
{∗} × G
π2
/
G × Gjj·jjjjj/ G
jj
jjjj
j
j
j
jjj
jjjj
j
j
j
jjj
jjjj
j
j
j
jjjj
jjjj
Id
G
• Right (left) inverse G
Id×·−1
{∗}
e
/
G×G
/
·
G
A Topological Group is a group G endowed with a
topology such that both multiplication and inversion
are continuous operations.
A Compact Group is a topological group such that it
is compact as a topological space, namely every open
cover of G admits a finite subcover
A group G with the discrete topology is compact if and
only if it is finite.
Examples of Finite Groups
• The Symmetric and Alternating groups Sn, An.
• The Dihedral groups Dn
• The unit quaternion group Q8.
• The group of automorphisms of a finite dimensional
vector space over a finite field: GLn(Fq )
Examples of compact groups
• The groups of rotations on a finite dimensional
Hilbert space over R: SO(n).
• The groups of Unitary transformations of a finite
dimensional Hilbert space over C: SU(n).
• The torus T n.
• The Spin groups
1 → Z/2Z → Spin(n) → SO(n) → 1
Examples of non-compact groups
• The group of affine transformations of a real vector
space.
• The group of volume-preserving transformations of
a real vector space SL(n; R).
• The Heisenberg group of upper triangular 3 × 3 matrices with entries in the real numbers and 1 in the
diagonal.
• The group of homeomorphisms of the circle S 1.
A Lie Group is a topological group which also happens
to be a differentiable manifold and both multiplication
and inversion are smooth maps.
• The additive group of p-adic integers is a compact
group which is not a Lie group.
• Q with its topology inherited from R is a topological
group which is not a Lie group.
• The group of diffeomorphisms of the circle is a
topological group which is not a Lie group.
A finite dimensional representation of a group G is a
finite dimensional vector space V (over a field k) and
an homomorphism of groups ρ : G → GL(V ).
If G is a topological group we say that (V, ρ) is continuous if ρ is.
If G is a Lie group we say that (V, ρ) is smooth if ρ
is.this is vacuous
Alternatively we obtain an action map a : G × V → V ,
as (g, v) 7→ ρ(g)v.
G × G × V Id×a
/
G×V
a
·×Id
G×V
say something about the unit!
We say that G acts on V .
a
/
V
S4 acts on R3 as symmetries of the Cube.
Equivalently, the group of signed permutation 3 × 3
matrices (of determinant 1) is S4.
The group SU(2) acts on R3.
As a manifold (and thus as a topological space) the
group SU(2) is the three sphere S 3.
Haar Measure
Let G be a compact group. There exists a unique measure dt such that
R
R
1. G f (t)dt = G f (ts)dt for each continuous f and
each s ∈ G (invariance under right translation).
R
2. G dt = 1 (normalization).
It follows that dt is also invariant under left translations.
A representation ρ of G on a complex Hilbert space is
said to be unitary if every operator ρ(g) is unitary, that
is
(ρ(g)v, ρ(g)w) = (v, w)
∀v, w ∈ V,
g ∈ G.
Lemma Let G be a finite group and let V be a complex
finite dimensional representation. Then there exists an
Hermitian product (·, ·) on V such that the representation is Unitary with respect to that inner product.
Proof Start with any Hermitian product (·, ·)0 on V .
And define a new one (·, ·) by
1 X
(v, w) =
(ρ(g)v, ρ(g)w)
|G| g∈G
V ∈ G-mod. V G = {v ∈ V |ρ(G)v = v} are called invariant vectors.
1 X
Z=
ρ(g) :
|G| g∈G
V →VG
V̄ is the conjugate representation given by g 7→ ρ(g).
V, W ∈ G-mod ⇒ V ⊗ W ∈ G-mod
(, ) = Z(, )0 ∈ (V ⊗ V̄ )G
A morpshim of representations S : V → W is a linear
map such that
G×V
Id×S
/
G×W
aV
aW
V
S
/
W
Two representations are isomorphic if there exists S ∈
HomG(V, W ) which is invertible as a linear map.
Every finite dimensional complex representation of G is
isomorphic to a unitary representation.
GLL
ρ
L
L
L
L
L
∃
/
L
L
L
GL(V
)
O
L
S·S −1
L%
U (V )
V, W ∈ G-mod ⇒ V ⊕ W ∈ G-mod.
ρV ⊕W (g) =
ρV (g)
0
0
ρW (g)
!
A representation is said indecomposable if it is not
isomorphic to a direct sum with V and W not 0.
A subrepresentation of V is a subspace W ⊂ V such
that ρ(G)W ⊂ W
A representation V is said to be irreducible if its only
subrepresentations are 0 and V .
Lemma for a finite group G, irreducible and indecomposable representations are the same thing.
Lemma Every finite dimensional representation V of G
is isomophic to a direct sum of irreducibles.
Proof Let W ⊂ V be a subrepresentation of the Unitary
representation V . Then W ⊥ is a subrepresentation and
V ' W ⊕ W ⊥.
For a compact group G, the Hermitian product associated to (, )0 is
Z
G
(ρ(g)v, ρ(g)w)0dg =: (v, w)
Every finite dimensional representation is completely reducible.
Shur’s Lemma
(V, ρ) and (W, θ) two irreps of G. T ∈ HomG(V, W ) then
T = 0 or T is an isomorphism.
Proof KerT ⊂ V and ImT ⊂ V are submodules.
HomG(V, V ) ' C.
Proof If λ ∈ C is an eigenvalue of T , then T − λId is
not an iso.
Corollary Every irreducible representation of an Abelian
group is 1-dimensional.
For the group G = U (1) = {e2πiθ |θ ∈ R} ⊂ C we have
that the irreducible representations are parametrized by
n ∈ Z with
ρ(e2πiθ ) = e2πinθ
Schur orthogonality relations
(V, ρ) 6' (W, θ) two irreps. Then
Z G
Z G
ρ(g)v, v 0
ρ(g)v1, v10
0
θ(g)w, w dg = 0
∀v, v 0 ∈ V, w, w0 ∈ W
0 , v0 )
(v
,
v
)(v
1
2
1
2
ρ(g)v2, v20 dg =
dimV
∀v1, v10 , v2, v20 ∈ V
Proof G acts in Hom(W, V ) by conjugation T 7→ ρ(g)T θ(g)−1.
So that the projection to Hom(W, V )G is given by
L=
Z
G
ρ(g)T θ(g)−1dg
0 = HomG(W, V ) = Hom(W, V )G ⇐ W 6' V .
Choose T w0 = (w0, w)v and evaluate (Lw0, v 0) = 0.
For the second statement take the trace of L = λId.
Shur orhthogonality relations
Let i ∈ Ĝ parametrize irreps (Vi, ρi) of G.
dim Vi.
Let di =
{eij }1≤j≤di orthonormal basis of Vi.
√1 ρ(·)ei , ei
j k
di
is orthonormal in L2(G).
i∈I,1≤j,k≤di
A matrix coefficient of (V, ρ) is a function of the form
(ρ(·)v, w) ∈ C(G).
The character of (V, ρ) is the function
χV (g) = χρ(g) = TrV ρ(g) =
X
ρ(g)ej , ej
j
χρ is a Class function: χ(hgh−1) = χ(g).
χV ⊕W = χV + χW ,
χV̄ = χV ,
χV ⊗W = χV · χW
χHom(V,W ) = χW χV
Let (V, ρ) 6' (W, θ) be irreps of G.
Theorem (χρ, χρ) = 1 and (χρ, χθ ) = 0 in the L2 norm
of L2(G).
Theorem For arbitrary representations V, W . Then
χW , χ V
=
Z
G
χW χV = dim HomG(V, W )
Proof As before or just use that G ρ(g)dg : V → V G
R
therefore G χρ(g)dg = dim V G.
R
Peter-Weyl Theorem
The linear span of matrix coefficients all all irreducible
finite dimensional unitary representations of G is dense
in L2(G).
The linear span of characters of the above mentioned
representations is dense in the set of L2 class functions
on G.
Proof
• The closure U of the linear span of matrix coef. is
stable under f (x) 7→ f (x−1) and left/right translations.
• If U ⊥ 6= 0 it has a continuous function
1
−1 h)dg
lim
I
(g)H(g
N
−→ |N | G
Z
• In U ⊥ there’s continuous class function f , with f (x) =
f (x−1) and f (1) 6= 0.
Proof (cont)
• f 7→
R
F (g −1h)f (h)dh is Hilbert-Schidt compact.
• The finite dimensional eigenspace Vλ is a representation of G by left translations.
• Assume Vλ irred. {ei} an orthonormal basis, hi(g) =
(ρ(g)ei, ei) ∈ U .
(F, hi) = 0 ⇒ λ(ei, ei) = 0
Corollary
L2(G) =
M
⊕ dim Vi
=
Vi ⊗ Vi∗
Vi
i∈Ĝ
i∈Ĝ
M
Corollary
Every Compact finite dimensional Lie group is isomorphic to a group of matrices.
Proof Let G0 = G. Define Gi as follows. Choose
1 6= xi in the identity component of Gi−1. There exists
an irrep ρi ∈ Ĝ with ρi(xi) 6= 1. Define Gi = Ker ⊕j<i ρj .
Obtain ρ0 such that Kerρ0 = {y1, . . . , yn} is finite. Let
ρi be now such that ρ(yi) 6= 1. The representation
n
M
ρi
i=1
is faithful (one to one) and finite dimensional.
Lie algebras
A Lie algebra is a vector space g and a linear map
[, ] : g ∧ g → g such that the Jacobi condition holds
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0,
∀a, b, c ∈ g
gl(V ) = EndV with [S, T ] = S ◦ T − T ◦ S
Smooth vector fields X(M ) on a manifold M form a Lie
algebra.
• Let G be a Lie group. X ∈ X is called left invariant
if for every g ∈ G we have Lg∗X = X.
• Left invariant vector fields are closed under [, ]Lie so
they form a Lie algebra g0 = LieG
• g0 ' TeG hence it is a finite dimensional Lie algebra.
• g0 only depends on G0: the connected component
of e ∈ G.
The Adjoint representation
The group G acts on itself by conjugation by the map
c(g) : G → G given by h 7→ ghg −1.
The differential of this map at e produces a map G×g →
g which is an action.
Thus we define the representation Ad : G → GL(g) by
Ad(g) = c(g)∗,e.
V ∈ V ectC
dim V < ∞
X ∈ End(V ) ' gl(V )
t 7→ exp(tX) is a smooth map R → GL(V )
d exp(tX) = X
dt t=0
gl(V ) = Lie GL(V )
exp (Ad(g)X ) = g exp(X)g −1
det exp(X) = exp(trX)
sl(V ) = KerTr : gl(V ) → R
SL(V ) = Ker det : GL(V ) → GL(R) ' R∗
sl(V ) ' Lie SL(V )
(, ) : S 2V → R
SO(V ) := {X ∈ GL(V )|(Xv, Xw) = (v, w) ∀v, w ∈ V }∩SL(V )
so(V ) := {X ∈ gl(V )|(Xv, w) + (v, Xw) = 0 ∀v, w ∈ V }
so(V ) ' Lie SO(V )
(, ) : V ⊗ V → C Hermitian product.
SU (V ) := {X ∈ GL(V )|(Xv, Xw) = (v, w) ∀v, w ∈ V }∩SL(V )
su(V ) := {X ∈ gl(V )|(Xv, w) + (v, Xw) = 0 ∀v, w ∈ V }∩sl(V )
su(V ) ' Lie SU (V )
g modules
A Lie algebra homomorpshim ρ : g → gl(V ).
ρ [a, b] v = ρ(a)ρ(b)v − ρ(b)ρ(a)v
Homg(V, W ) = {T : V → W |ρW (a)T = T ρV (a) ∀a ∈ g}
Representations of G yield representations of g
ρ : G → GL(V )
ρ∗,e : g → gl(V )
Proof First see that ρ∗Ad(g) = Ad(ρ(g))ρ∗ and then
write g = exp(tX).
ρ exp = exp ρ∗
The reverse is much deeper
If G and H are analytic groups with G simply-connected
and ϕ : g → h is a morphism of their Lie algebras, then
there exists a group homomorphism Φ : G → H with
Φ∗,e = ϕ.
Representations of sl2
sl2 has basis e, f, h with
[h, e] = 2e,
[h, f ] = −2f,
[e, f ] = h
Vn = homogeneous polynomials of degree n in two variables x, y.
Theorem For each n ≥ 0 there exits a unique (up to
isomorphism) representation of sl2 of dimension n + 1.
It is given by Vn where
ρ(h) = x∂x − y∂y ,
ρ(e) = x∂y ,
ρ(f ) = y∂x
These are the unique finite dimensional irreducible representations of sl2.
Representations of G := SU (2)
As a topological space SU (2) ' S 3 ' SP (1) [It acts on
C2 preserving S 3 ⊂ C2]
(
U (1) ' S 1 ' T :=
eiθ
0
0
e−iθ
!)
⊂ SU (2) =: G
Hopf Fibration: S 1 ,→ S 3 → S 2
G/T ' S 2
The Flag Variety
The quotient G/T has a natural complex structure:
G ⊂ GC := SL(2, C)
(
N :=
!)
1 ∗
0 1
(
⊂ B :=
!)
∗ ∗
0 ∗
G → GC/B
G/T ' GC/B ' CP 1
⊂ GC
Representations of G := SU (2) (cont.)
For each n ≥ 0 there is a holomorphic line bundle O(n)
over G/T ' CP 1.
The group G acts on O(n) by bundle automorphisms.
To construct O(n) we start with a 1 dimensional representation of
C∗ ' TC :=
(
α 0
0 α−1
!)
⊂B
given by ρ(α) = αn.
Extend this representation to B as B/N = TC. Take
the associated bundle to the fibration
B ,→ GC → GC/B ' G/T
Representations of G := SU (2) (cont.)
Since G acts on O(n) compatible with the action on
G/T we obtain a representation Vn on its space of holomorphic sections!
Vn := H 0(G/T, O(n))
Concretely, if [x : y] are homogeneous coordinates on
CP 1, then holomorphic sections of O(n) are homogeneous polynomials of degree n in two variables x, y.
Back to su2
The action of the corresponding Lie algebra g0 := su2 or
rather its complexification g := sl(2, C) is via the global
holomorphic vector fields on CP 1, namely
h = x∂x − y∂y ,
e = x∂y ,
f = y∂x
The tangent bundle of CP 1 is isomorphic to O(2).
The adjoint representation of sl2 is therefore isomorphic
to V2.
An Ideal of g is a subspace I with [g, I] ⊂ I.
g = g0 ⊃ g1 = [g0, g0] ⊃ · · · ⊃ gi+1 = [gi, gi] ⊃ . . .
g is solvable if the ideals gj vanish for some j.
g = g0 ⊃ g1 = [g0, g] ⊃ · · · ⊃ gi+1 = [gi, g] ⊃ . . .
g is nilpotent if the ideals gj vanish for some j
Lemma ∃!r solvable ideal containing all other solvable
ideals of g. It’s called the radical of g and g is called
semisimple if r = 0.
g is simple if it has no non-zero proper ideals (and it’s
not Abelian)
Solvable and nilpotent Lie algebras

∗
0

b=
0
0
∗
∗
0
0
∗
∗
∗
0


∗
0
0
∗


 ⊃ n=
0
∗
∗
0
∗
0
0
0
∗
∗
0
0

∗
∗


∗
0
Root spaces for g = sln
h = diag(h1, . . . , hn) ⊂ g
χi ∈ h∗
χi(h) = hi.
We let eij be the matrix with non-vanishing i, j entry 1
for i 6= j.
ad(h)eij = (χi − χj )(h)eij
g=
M
λ∈h∗
gλ ,
ad(h)|gλ = λ(h)Id ∀h ∈ h
The sum is finite and the non-zero λ appearing in the
sum are called roots.
n
o
∆ = χi − χj
1≤i,j≤n



= gα+β
[gα, gβ ] ⊂ h



=0
if α + β is a root
if α + β = 0
otherwise
We can consider ∆ ⊂ h∗0 where h0 are the diagonal
matrices with Real entries.
∆ = ∆+ t ∆−
α ∈ ∆+ ⇔ −α ∈ ∆−
There exists a largest root θ ∈ ∆+.
sln is simple
Let a ⊂ g be an ideal.
If a ⊂ h then ∃α ∈ ∆, 0 6= a ∈ gα such that 0 6= gα =
[a, a] ⊂ a, contradiction.
Let a = h +
with aα 6= 0.
P
aα ∈ a. Suppose β < 0 is the smallest α
There exists γ, δ ∈ ∆ such that β+γ ∈ ∆ and β+γ +δ =
θ.
[gδ , [gγ , a]] = gθ ⊂ a
A Root system is a pair (V, ∆) where
V is a finite dimensional real vector space with a symmetric positive definite bilinear form (, ).
∆ ⊂ V is a finite set such that
1. 0 ∈
/ ∆, ∆ spans V over R.
2. α ∈ ∆ ⇒ (kα ∈ ∆ ⇔ k = ±1)
3. α, β ∈ ∆, the set {β + jα} ∩ (∆ ∪ {0}) is a string
where j ∈ [−q, p] ⊂ Z and p − q = 2(α, β)(α, α)−1.
Let (V, ∆) be a root system.
g0 := h := V ⊗R C,
gα := Ceα, α ∈ ∆
M
g :=
gα
α∈∆∪{0}
Is a Lie algebra such that
[h, eα] = α(h)eα,



= gα+β
[gα, gβ ] ∈ h



=0
if α + β ∈ ∆
if α + β = 0
otherwise
For each α ∈ ∆ we have that gα ⊕ g−α ⊕ [gα, g−α] ' sl2.
All simple Lie algebras over C are built from sl2.