Group Representations III

Dr Timothy Murphy

Joly Theatre

Friday, 25 April 2003

15:15–16:45

Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions carry the same number of marks.

In this exam, ‘Lie algebra’ means Lie algebra over R, and ‘representation’ means finite-dimensional representation over C.

1. Define the exponential eX of a square matrix X, and show that eX is invertible. Determine eX in each of the following cases:

X=

X=

0

0

1

0

1

, X=

0

1

, X=

1

0

1

1

1

1

, X=

0

1

, X=

2

0 −1

,

1 0

−1 0

.

0 −1

Which of these 6 matrices X are themselves expressible in the form

X = eY , (a) with a real matrix Y , (b) with a complex matrix Y ?

(Justify your answers in all cases.)

2. Define a Lie algebra.

Show that there are just two Lie algebras over R, up to isomorphism; and give two linear groups having these as their Lie algebras.

Continued overleaf

3. Define the dimension of a linear group; and determine the dimensions of each of the following groups:

O(n), SO(n), U(n), SU(n), GL(n, R),

SL(n, R), GL(n, C), SL(n, C), Sp(n), E(n)?

(E(n) is the isometry group of n-dimensional Euclidean space.)

4. Determine the Lie algebras of SU(2) and SO(3), and show that they are isomomorphic.

Show that the 2 groups themselves are not isomorphic.

5. Determine the Lie algebra of SL(2, R), and find all the simple representations of this algebra.

Do these representations all arise from representations of the group

SL(2, R)?

6. What is meant