Course 424

Group Representations III

Dr Timothy Murphy

Arts Block 3051 Thursday, 17 June 1993 14:00–16:00

Answer as many questions as you can^{1}; 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 & -1 \end{matrix} ) , X = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}0 & -1 \\ 1 & 0 \end{matrix} ) , X = ( \begin{matrix}1 & -1 \\ 1 & 1 \end{matrix} ) , X = ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} )

Show that if X has eigenvalues λ, μ then e^{X} has eigenvalues e^{λ}, e^{μ}.

Which of the above 5 matrices X are themselves expressible in the form X = e^{Y} for some real matrix Y? (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.

1Nb: There is a question overleaf.

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)?

3. 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.

4. Define a representation of a Lie algebra ℒ. What is meant by saying that such a representation is (a) simple, (b) semisimple?

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

Show that every representation of the group SL(2, ℝ) is semisimple, stating carefully but without proof any results you need.

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.

Show that the groups 𝕋^{m} × ℝ^{n} and 𝕋^{m′} × ℝ^{n′} are isomorphic if and only if m = m′ and n = n′.