8/13/2019 Lie 2006 Exam

1/4

Kings College LondonUniversity Of London

This paper is part of an examination of the College counting towards the award of a degree.

Examinations are governed by the College Regulations under the authority of the AcademicBoard.

ATTACH this paper to your script USING THE STRING PROVIDED

Candidate No: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Desk No: . . . . . . . . . . . . . . . . . . . . . . .

MSci Examination

CMMS01/CM424Z Lie groups and Lie algebras

Summer 2006

Time Allowed: Two Hours

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

Section A contributes about 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

2006 cKings College London

8/13/2019 Lie 2006 Exam

2/4

- 2 - Section A CMMS01/CM424Z

SECTION A

A 1. (14 points)

(i) State the definition of the Killing form of a finite-dimensional Lie algebra.

(ii) Let g be a finite-dimensional Lie algebra and let g be its Killing form.

Show that g([x, y], z) = g(x, [y, z]) for all x,y,z g. (You may use therepresentation property of the adjoint action without proof.)

(iii) Show that ifg is abelian, then g(x, y) = 0 for all x, yg .

A 2. (20 points)

(i) State the definition of the properties simple and semi-simple of a Lie

algebra.

(ii) Show that ifg is simple, then [g, g] =g where [g, g] = span

[x, y] |x, yg.Hint: Show that [g, g] is an ideal.

(iii) Recall the definition of the adjoint action adx(y) = [x, y]. Show that for

a simple Lie algebra g, there is no nonzero x g such that adx : g g isidentically zero.

A 3. (16 points)

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

Cartan subalgebra.

(i) Give the formula for the Weyl reflection s with respect to a root ofg.

(ii) Show

that s() =, and that (, ) = 0 (for some h) implies s() =.

(iii) Show that det(s) =1.Hint: Find a basis ofh consisting of and vectors uk such that (, uk) = 0.

See Next Page

8/13/2019 Lie 2006 Exam

3/4

- 3 - Section B CMMS01/CM424Z

SECTION B

B 4. (i) Letn1. Describe the complex Lie algebrasgl(n,C) andsl(n,C) as subsetsof Mat(n,C). Give the dimensions ofgl(n,C) and sl(n,C) (over C).

(ii) Give the Lie bracket ofgl(n,C) andsl(n,C) and show that for two matrices

x, ysl(n,C) also [x, y]sl(n,C).(iii) Show that sl(n,C) is an ideal ofgl(n,C).

(iv) Denote by C the one-dimensional abelian complex Lie algebra (i.e. C is

equipped with the Lie bracket [, ] = 0 for all, C).Show that sl(n,C)

C

=gl(n,C) as complex Lie algebras.

Hint: Consider the linear map (M) =M+ 1fromsl(n,C)Cto gl(n,C)(here 1 is the nn unit matrix).

B 5. (i) State the definition of a matrix Lie group.

(ii) Show that the set of matrices U(n) ={M Mat(n,C) |MM=1}, where1 is the n

n-unit matrix, is a matrix Lie group (and in particular that it is a

group).

(iii) State the definition of the Lie algebra of a matrix Lie group.

(iv) Find the Lie algebrau(n) ofU(n) and prove your answer. (You may assume

the various properties of the matrix exponential.)

(v) Verify that for x, yu(n) also [x, y]u(n).

See Next Page

8/13/2019 Lie 2006 Exam

4/4

- 4 - Section B CMMS01/CM424Z

B 6. Let gbe a finite-dimensional complex semi-simple Lie algebra of rank 2. Suppose

g has the following root diagram:

i.e. the angle between neighbouring roots is 45 and their length-ratio is

2.

(i) What is the dimension ofg? Give a short explanation of your answer.

(ii) Describe briefly the procedure to find a choice of positive roots and a set of

simple roots. Apply this procedure to the root diagram above. (Copy the root

diagram to your answer sheet and put labels accordingly.)

(iii) State how the Cartan matrix Aij is defined in terms of simple roots (i).

Compute the Cartan matrix resulting form the simple roots found in part (ii).

(iv) Draw the Dynkin diagram ofg.

B 7. The Virasoro algebra is the infinite-dimensional complex Lie algebra V with

basis{C} {Lm |m Z} and Lie bracket on the basis elements given by[C, C] = [C, Lm] = [Lm, C] = 0 ,

[Lm, Ln] = (mn) Lm+n+ 112m+n,0(m3m) C .(i) Show that the above Lie bracket is indeed skew-symmetric. Verify the Jacobi

identity for the three elements L1, L2, L3 (i.e. check [L1, [L2, L3]] + = 0).

(ii) Letb : VVC

be an invariant, symmetric, bilinear form (i.e. bis bilinearand b(x, y) =b(y, x) as well as b([x, y], z) =b(x, [y, z]) for all x, y, z V.) Showthat b is degenerate (i.e. there exists an x V such that b(x, y) = 0 for allyV).Hint: One may proceed as follows:

Use [Lm, L0] =mLm to see that b(Lm, C) = 0 for all m= 0. Use [Lm, Lm] = 2mL0 + 112(m3 m)C in b( , C) to conclude that

b(C, C) = 0 and b(L0, C) = 0.

Final Page