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MSc Examination

7CCMMS01 (CMMS01) Lie groups and Lie algebras

Summer 2008

Time Allowed: Two Hours

This paper consists of two sections, Section A and Section B.

Section A contributes half the total marks for the paper.

Answer all questions in Section A.

All questions in Section B carry equal marks, but if more than two are

attempted, then only the best two will count.

NO calculators are permitted.

TURN OVER WHEN INSTRUCTED

2008 c©King’s College London

- 2 - Section A 7CCMMS01

SECTION A

A1. (22 points)

(i) State the definition of a Lie algebra over the complex numbers C.

(ii) State the definition of an ideal of a Lie algebra.

(iii) Let U be the Lie algebra of 2×2 upper triangular matrices

U ={(

a b

0 c

) ∣∣∣ a, b, c ∈ C}

with matrix commutator as Lie bracket. Show that

I ={(

0 d

0 0

) ∣∣∣ d ∈ C}

is an ideal in U . (You do not need to prove that U is a Lie algebra.)

(iv) For U as in (iii), show that

N ={(

d 0

0 0

) ∣∣∣ d ∈ C}

is not an ideal in U .

(v) State the definition of a Lie algebra homomorphism ϕ : g → h between two

Lie algebras g and h.

(vi) Let U be as in (iii) and N as in (iv). Show that there cannot be a Lie

algebra h and a Lie algebra homomorphism ψ : U → h such that ker(ψ) = N .

Hint: Prove that the kernel ker(ϕ) of a Lie algebra homomorphism ϕ : g → h

is an ideal of g.

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- 3 - Section A 7CCMMS01

A2. (14 points)

Let g be a finite dimensional complex Lie algebra.

(i) State the definition of the Killing form κ of g

(ii) Show that κ([x, y], z) = κ(x, [y, z]) for all x, y, z ∈ g. (You may use the

representation property of the adjoint action without proof.)

(iii) Let h be an ideal in g and define

h⊥ = {x ∈ g |κ(x, b) = 0 for all b ∈ h } .

Show that also h⊥ is an ideal in g.

A3. (14 points)

Let g be a finite dimensional simple complex Lie algebra with Cartan subalgebra

h and Cartan matrix

A =

2 −1 0 0

−1 2 −1 −1

0 −1 2 0

0 −1 0 2

(i) State the definition of the Cartan matrix Aij via the simple roots α(i).

(ii) Draw the Dynkin diagram for g.

(iii) State the formula for the Weyl reflection sα with respect to a root α of g.

(iv) Show that sα(1)(α(2)) = α(1) + α(2) for the simple roots α(1) and α(2) of g.

Is α(1) + α(2) a root of g? Briefly state the property you used to answer this

question. (The ordering of the simple roots is the same as that of the rows or

columns in A.)

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- 4 - Section B 7CCMMS01

SECTION B

B4. Let

Jsp =

(0 1n×n

−1n×n 0

)∈ Mat(2n,R) .

and define SP (2n) = {M ∈ Mat(2n,R) |M tJspM = Jsp }.

(i) State the definition of a matrix Lie group.

(ii) Show that if M ∈ SP (2n) then det(M) ∈ {±1}.Hint: Show that (Jsp)

2 = −12n×2n, so that in particular det(Jsp) 6= 0.

(iii) Show that SP (2n) is a matrix Lie group (and thus in particular that it is

closed under taking inverses and under multiplication).

(iv) Consider the special case n = 1. Show that for M ∈ SP (2) one always has

det(M) = 1 (and never det(M) = −1).

Hint: Explicitly write out the condition M tJspM = Jsp for the 2×2-matrix M .

B5. Let g be a complex Lie algebra of finite dimension n = dim(g). Suppose that

g has a basis {T a | a = 1, . . . , n } such that κ(T a, T b) = δa,b. Recall that the

structure constants fabc of g are defined by [T a, T b] =∑n

c=1 fabc Tc.

(i) Show that fabc is anti-symmetric in all three indices. (You may use the

properties of the Killing form κ without proof.)

Hint: Consider κ([T a, T b], T c).

(ii) State the definition of a representation of a Lie algebra.

(iii) Let (V,R) be a representation of g. Define the linear map C : V → V as

C =n∑

a=1

R(T a)R(T a) .

Show that R(T b)C = CR(T b) for all b = 1, . . . , n.

Hint: Prove first that R(T b)R(T a) = R(T a)R(T b) +∑n

c=1 fbacR(T c). Apply

this twice and use part (i).

(iv) Let (V,R) and C be as in (iii) and suppose in addition that (V,R) is

finite-dimensional and irreducible. Prove that C = λ idV , where λ ∈ C is a

constant, and idV is the identity map on V . (You may use Schur’s lemma and

its consequences without proof.)

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- 5 - Section B 7CCMMS01

B6. Let g be a finite dimensional semi-simple complex Lie algebra and let h be a

Cartan subalgebra of g. For ϕ ∈ h∗ let

gϕ = {x ∈ g | [H, x] = ϕ(H)x for all H ∈ h } .

(i) State the definition of a root of g.

(ii) Let λ, µ ∈ h∗, and let x ∈ gλ and y ∈ gµ (possibly both zero). Show that

[x, y] ∈ gλ+µ.

(iii) With the notation of (ii), show that if λ+ µ 6= 0 then κ(x, y) = 0.

Hint: Use the invariance of κ from question A2 (ii).

For the following two questions you may use all results of the lecture without

proof.

(iv) Recall that the rank r of g is given by r = dim(h). Show that dim(g) ≥ 3r.

Hint: The roots span the vector space h∗.

(v) Show that there is no semi-simple complex Lie algebra of dimension 1,2,4

or 5.

Hint: A finite-dimensional semi-simple complex Lie algebra of rank 1 is isomor-

phic to sl(2,C).

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