Course 424
Group Representations III
Dr Timothy Murphy
Sam Beckett Theatre Wednesday, 10 June 1991 14:00–16:00
Answer as many questions as you can; all carry the same number of marks.
Unless otherwise stated, all Lie algebras are over R, and all representations are finite-dimensional over C.
1. Define the exponential eX of a square matrix X.
Determine eX in each of the following cases:
X=

1
0

0
2

,

X=

0
1

1
0

,

X=

0
1

-1
0

,

X=

1
0

0
-1

Which of these 5 matrices X are themselves expressible in the form
X = eY , with (a) Y real, (b) Y complex? (Justify your answers in all cases.) 2. Define a linear group, and a Lie algebra; and define the Lie algebra L G of a linear group G, showing that it is indeed a Lie algebra.
Determine the Lie algebras of SU(2) and SO(3), and show that they are isomomorphic.

,

X=

1
0

1

3. Define a representation of a Lie algebra; and show how each representation α of a linear group G gives rise to a representation L α of
L G.
Determine the Lie algebra of SL(2, R); and show that this Lie algebra sl(2, R) has just 1 simple representation of each dimension 1, 2, 3, . . ..
4. What is meant by saying that a connected linear group G is simplyconnected? Show that SU(2) is simply-connected.
Sketch the proof that if the linear group G is connected and simplyconnected then every representation of L G lifts to a representation of
G.
Show that if 2 real Lie algebras have the same complexification then their representations (over C) correspond.