THE THEOREM OF THE HIGHEST WEIGHT The Theorem of the ...

THE THEOREM OF THE HIGHEST WEIGHT
ANKE D. POHL
Abstract. Incomplete notes of the talk in the IRTG Student Seminar
07.06.06. This is a draft version and thought for internal use only.
The Theorem of the Highest Weight states that the equivalence classes of
irreducible finite-dimensional representations of a finite-dimensional complex
semisimple Lie algebra g can be characterized by their highest weight.
Moreover, it tells that the set of highest weight are identical with the set of
dominant algebraically integral functionals on a Cartan subalgebra of g.
In the first section we collect the definitions and propositions we need for
the statement and (partial) proof of the Theorem of the Highest Weight.
The omission of proofs does not mean that these are trivial. All proofs can
be found in the book [Kna02] on which all of the talk is heavily based. The
second section contains a statement of the Theorem of the Highest Weight.
We will prove the stated properties of the highest weight and that the correspondence
is injective. The proof of the surjectivity of the correspondence
is very long hard work and can be found in [Kna02].
1. Preliminaries
A complex Lie algebra g is a complex algebra such that the C-bilinear
product [·, ·]: g×g → g satisfies the following two properties for all X, Y, Z ∈
g
(a) antisymmetry: [X, Y ] = −[Y, X],
(b) Jacobi identity: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [Y, X]] = 0.
Note that the antisymmetry implies that g is abelian if and only if [X, Y ] = 0
for all X, Y ∈ g.
If V is a complex vector space, then End C (V ) endowed with the product
[X, Y ] := X ◦ Y − Y ◦ X is a complex Lie algebra.
In what follows, g will always denote a complex Lie algebra and V a complex
vector space.
A representation π of g on V is a Lie algebra homomorphism π : g →
End C (V ). An invariant subspace for such a representation is a vector
subspace U of V such that π(g)U ⊆ U. A representation is called irreducible
if the only invariant subspaces are 0 and V .
1

2 A. POHL
Let π be a representation of g on V , and π ′ a representation of g on V ′ .
Then π and π ′ are called equivalent if there is a vector space isomorphism
E : V → V ′ such that E ◦ π(X) = π ′ (X) ◦ E for all X ∈ g.
The equivalence classes of irreducible finite-dimensional representations of
g are denoted by ĝ fin .
We need to define three subclasses of Lie algebras: the solvable, the semisimple
and the nilpotent ones. If a and b are subsets of g, then
[a, b] := 〈{[X, Y ] | X ∈ a, Y ∈ b}〉
denotes the Lie subalgebra of g spanned by the elements [X, Y ].
For n ∈ N we define g 0 := g and g n+1 := [g n , g n ]. The commutator series
of g is the series
g 0 ⊇ g 1 ⊇ g 2 ⊇ . . . .
If there is an n ∈ N 0 such that g n = 0, then g is called solvable.
The radical rad g is the largest solvable ideal in g, its existence can be
proved for each Lie algebra g (cf. [Kna02, Prop 1.12]). Finally, g is called
semisimple, if g has no nonzero solvable ideals, that is to say, rad g = 0.
Next we define for n ∈ N the ideals g 0 := g and g n+1 := [g, g n ]. The series
g 0 ⊇ g 1 ⊇ g 2 ⊇ . . .
is called the lower central series for g. If there is an n ∈ N 0 such that
g n = 0, then g is called nilpotent.
Recall that the (infinitesimal) adjoint representation ad g : g → End C (g) is a
representation of g on itself.
Proposition 1.1 ([Kna02, Prop 2.5]). Let g be a finite-dimensional complex
Lie algebra and h a nilpotent Lie subalgebra. For α ∈ h ∗ = L(h, C) define
Then we have
g α := {X ∈ g | ∀ H ∈ h ∃ n ∈ N: (ad g H − α(H)1) n X = 0}.
(a) g = ⊕ α∈h ∗ g α , (b) h ⊆ g 0 , (c) [g α , g β ] ⊆ g α+β .
If g α ≠ 0, then g α is called a generalized weight space of g relative
to ad g h. The item (c) shows that g 0 is a subalgebra. A nilpotent Lie
subalgebra h of g is called a Cartan subalgebra if h = g 0 .
Of special importance is the next proposition. In general, for an infinitedimensional
or real Lie algebra a Cartan subalgebra does not exist.
Proposition 1.2 ([Kna02, Thm 2.9, Thm 2.15]). Any finite-dimensional
complex Lie algebra g has a Cartan subalgebra, and this is unique up to
(inner) automorphism.

THEOREM OF THE HIGHEST WEIGHT 3
If h is a Cartan subalgebra of g, then the generalized weight spaces of Proposition
1.1 are called root spaces, the nonzero weights are called roots, and
∆ = ∆(g, h) denotes the (finite) set of roots. The decomposition
g = h ⊕ ⊕ α∈∆
g α
is known as the root-space decomposition of g with respect to h. Elements
of g α are called root vectors for the root α.
If g is semisimple, then a Cartan subalgebra has special properties and the
root spaces have a simple form.
Proposition 1.3 ([Kna02, Prop 2.10, Prop 2.13, Cor 2.23, Prop 2.21]). Let
g be a complex semisimple Lie algebra and h a Cartan subalgebra. Then h is
abelian and ad g (h) is simultaneously diagonable. For each α ∈ ∆, the root
space g α is one-dimensional and
g α = {X ∈ g | ∀ H ∈ h: ad(H)X = α(H)X}.
From now on let g be a complex semisimple finite-dimensional Lie algebra
and h a Cartan subalgebra. Let π be a representation of g on V . For
λ ∈ h ∗ = L(h, C) (complex linear functionals) we put
V λ := {v ∈ V | ∀ H ∈ h ∃ n ∈ N: (π(H) − λ(H)1) n v = 0}.
If V λ ≠ 0, then V λ is called a generalized weight space, the elements of
V λ are called generalized weight vectors, and λ is a weight.
The weight space belonging to a weight λ is
{v ∈ V | ∀H ∈ h: π(H)v = λ(H)v},
which is obviously a subspace of V λ . Elements of the weight space are called
weight vectors.
For X, Y ∈ g we define B(X, Y ) := Tr(ad X ◦ ad Y ). Then B is a symmetric
bilinear form on g which is called the Killing form. It has a couple of
properties of which the following is important for us.
Proposition 1.4 ([Kna02, Prop 2.17]). B| h×h is nondegenerate. Therefore
to each element α ∈ h ∗ corresponds a unique H α ∈ h such that
α(H) = B(H, H α ) for all H ∈ h.
The Killing form B allows to define a dual bilinear form on h ∗ via
〈ϕ, ψ〉 := B(H ϕ , H ψ ) for ϕ, ψ ∈ h ∗ .
A real Lie algebra g 0 is said to be a real form of g if
g = g 0 ⊕ ig 0
as real vector spaces. The next proposition tells us how to construct a real
Lie algebra h 0 that is a real form of h such that h ∗ 0 = L(h 0, R) (real linear

4 A. POHL
functionals) is the real form of h ∗ on which all the roots are real valued. The
proof uses special properties of the Killing form.
Proposition 1.5 ([Kna02, Cor 2.38]). Let W be the R linear span of ∆ in
h ∗ . Then W is a real form of the vector space h ∗ , and the restriction of the
bilinear form 〈·|·〉 to W × W is a positive-definite inner product. Moreover,
if h 0 denotes the R linear span of all H α for α ∈ ∆, then h 0 is a real form
of the vector space h, the members of W are exactly those linear functionals
that are real on h 0 , and restriction of the operation of those linear functionals
from h to h 0 is an R isomorphism of W onto h ∗ 0 .
Note that the previous proposition shows that the Killing form is positive
definite on h 0 .
Three notions are absolutely important in the Theorem of Highest Weight.
The first is, of course, “highest weight” which is defined via a total ordering
on h ∗ 0 . The second is “algebraically integral functional”, and the third
is “dominant functional”. In fact, the Theorem of Highest Weight states
(among more important things) that the highest weight is an algebraically
integral dominant weight, and viceversa that every algebraically integral
dominant weight occurs as a highest weight.
An element λ of h ∗ is called algebraically integral if 2〈λ, α〉/|α| 2 is an
integer for each α ∈ ∆.
Total ordering on h ∗ 0 : The aim is to define a subset of h∗ 0 to be the set of
positive elements such that the notion of positivity on h ∗ 0 meets the following
two properties:
(P1) For each nonzero ϕ ∈ h ∗ 0 , exactly one of ϕ and −ϕ is positive,
(P2) the sum of positive elements is positive, and any positive multiple of
a positive element is positive.
Then we define that ϕ > ψ or ψ < ϕ if ϕ − ψ is positive. This way a total
ordering of h ∗ 0 is established. If α ∈ h∗ 0 is a root, it is called simple if α > 0
and if α does not decompose as α = β 1 + β 2 where β 1 , β 2 are both positive
roots. The set of positive roots is denoted ∆ + , and the subset of simple
roots is denoted Π. An element λ of h ∗ is called dominant if 〈λ, α〉 ≥ 0 for
all α ∈ ∆ + .
One way to define such a notion of positivity is via a lexicographic ordering.
One fixes a finite spanning set H 1 , . . . , H m of h 0 as (real) vector
space, e.g. a basis, and endows it with a total ordering, here 1, . . . , m. Then
one defines ϕ ∈ h ∗ 0 to be positive if there exists a k ∈ {1, . . . , m} such that
ϕ(H i ) = 0 for 1 ≤ i ≤ k − 1 and ϕ(H k ) > 0.
Proposition 1.8 states that each weight can be regarded as an element of
h ∗ 0 . The largest weight w.r.t. to the introduced ordering of h∗ 0 is called the
highest weight. Hence, a priori, the notion of “highest weight” heavily
depends on the ordering.

THEOREM OF THE HIGHEST WEIGHT 5
Proposition 1.6 ([Kna02, Prop 2.49]). If l = dim R h ∗ 0 , then there are l
simple roots α 1 , . . . , α l , and they are linearly independent. If β is a root and
is written as β = x 1 α 1 + · · · + x l α l , then all the x j have the same sign (0 is
considered to be positive or negative, as needed), and all the x j are integers.
The proof that each highest weight is dominant is based on the fact that g
is spanned by embedded copies of sl(2, C).
The Lie algebra sl(2, C) is generated by the vector space basis
( ) ( ) ( )
1 0
0 1
0 0
h = , e = , f =
0 −1 0 0
1 0
as a Lie subalgebra of the Lie algebra End C (C 2 ). The bracket relations are
[h, e] = 2e, [h, f] = −2f, [e, f] = h.
Let α be a root and let H α ∈ h be the vector defined in Proposition 1.4.
Select nonzero elements E α ∈ g α and E −α ∈ g −α . Then [Kna02, Cor 2.25]
states that the pair {E α , E −α } can (and should) be normalized such that
B(E α , E −α ) = 1. If we set
H α ′ := 2
α(H α ) H α, E α ′ := 2
α(H α ) E α, E −α ′ := E −α,
then we have the bracket relations
[H ′ α, E ′ α] = 2E ′ α, [H ′ α, E ′ −α] = −2E ′ −α, [E ′ α, E ′ −α] = H ′ α.
Hence {H ′ α, E ′ α, E ′ −α} spans a Lie aubalgebra sl α of g isomorphic to sl(2, C).
The next proposition will guarantee that each highest weight is dominant.
Proposition 1.7 ([Kna02, Cor 1.72]). Let π be a complex-linear representation
of sl(2, C) on a finite-dimensional complex vector space V . Then
π(h) is diagonable, all its eigenvalues are integers, and the multiplicity of
an eigenvalue k equals the multiplicity of −k.
The following proposition is crucial in the proof of the Theorem of the
Highest Weight.
Proposition 1.8 ([Kna02, Prop 5.4]). Let h 0 be the real form of h. If π is
a representation of g on the finite-dimensional complex vector space V , then
(a) π(h) acts diagonably on V , so that every generalized weight vector is a
weight vector and V is the direct sum of all the weight spaces,
(b) every weight is real valued on h 0 and is algebraically integral,
(c) roots and weights are related by π(g α )V λ ⊆ V λ+α and π(h)V λ ⊆ V λ .
The universal enveloping algebra allows to extend a representation ϕ: g →
End C (V ) uniquely to a homomorphism of algebras U(g) → End C (V ). This
extension is the right framework for investigations on iterated actions of g
on V .

6 A. POHL
Proposition 1.9 ([Kna02, Prop 3.3, Remarks 2)]). Let g be a complex Lie
algebra. There is a pair (U(g), ι) such that U(g) is an associative complex
algebra with unit and ι: g → U(g) is a linear mapping satisfying
ι[X, Y ] = ι(X)ι(Y ) − ι(Y )ι(X)
for all X, Y ∈ g
such that (U(g), ι) meets the following universal property: Whenever A is
an associative complex algebra with unit and π : g → A is a linear mapping
satisfying
π[X, Y ] = π(X)π(Y ) − π(Y )π(X)
for all X, Y ∈ g
then there exists a unique algebra homomorphism ˜π : U(g) → A such that
˜π(1) = 1 and the diagram
commutes.
ι
g
U(g)
π
As usual, one proves that U(g) is unique up to isomorphism and that ι is
injective, therefore we will identify X and ι(X) for each X ∈ g. The algebra
U(g) is called the universal enveloping algebra of g.
The Poincaré-Birkhoff-Witt Theorem gives a vector space basis for U(g).
Proposition 1.10 (Poincaré-Birkhoff-Witt, [Kna02, Thm 3.8]). Let {X i } i∈A
be a basis of g, and suppose a total ordering has been imposed on the index
set A. Then the set of all monomials
˜π
X j 1
i 1 · · · X jn
i n
,
where i 1 < · · · < i n and all j k ≥ 0, is a basis of U(g).
The following version of Schur’s Lemma is needed in the proof that the correspondence
that is stated in the Theorem of the Highest Weight is injective.
Proposition 1.11 (Schur’s Lemma, [Kna02, Prop 5.1]). Suppose ϕ and
ϕ ′ are irreducible representations of a finite-dimensional Lie algebra g on
finite-dimensional vector spaces V and V ′ , resp. If L: V → V ′ is a linear
map such that ϕ ′ (X)L = Lϕ(X) for all X ∈ g, then L is bijective or L = 0.
A
2. The Theorem of the Highest Weight
Let g be a complex semisimple Lie algebra, let h be a Cartan subalgebra, let
∆ = ∆(g, h) be the set of roots, and let W (∆) be the Weyl group. Let B be
the Killing form of g and let h 0 be the real form of h on which all the roots
are real valued. Introduce an ordering in h ∗ 0 . Let Π denote the resulting set
of simple roots, and ∆ + that one of positive roots.

THEOREM OF THE HIGHEST WEIGHT 7
Theorem 2.1 (Theorem of the Highest Weight (algebraic), [Kna02, Thm
5.5]). The map
ĝ fin → {dominant algebraically integral linear functionals on h}
[π] ↦→ highest weight of π
is a bijective correspondence. The highest weight λ of π λ has the additional
properties:
(a) λ depends only on the simple system Π and not on the ordering used to
define Π,
(b) the weight space V λ for λ is one-dimensional,
(c) each root vector E α for arbitrary α ∈ ∆ + annihilates the members of
V λ , and the members of V λ are the only vectors with this property,
(d) every weight of π is of the form λ − ∑ l
i=1 n iα i with the integers n i ≥ 0
and the α i ∈ Π.
References
[Kna02] Anthony W. Knapp, Lie groups beyond an introduction, second ed., Progress in
Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002.