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 e^{X} of a square matrix X.

Determine e^{X} in each of the following cases:

X = ( \begin{matrix}1 & 0 \\ 0 & 2 \end{matrix} ) , X = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}0 & -1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}1 & 0 \\ 0 & -1 \end{matrix} ) , X = ( \begin{matrix}1 & -1 \\ 0 & 1 \end{matrix} )

Which of these 5 matrices X are themselves expressible in the form X = e^{Y}, 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.

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. Hence or otherwise show that all the representations of sl(2, ℝ) are semisimple.

5. Show that every connected abelian linear group A is isomorphic to

𝕋^{m} × ℝ^{n}

for some m and n, where 𝕋 denotes the torus ℝ/ℤ

Express the multiplicative group ℂ^{×} in this form.