Course 424

Group Representations III

Dr Timothy Murphy

School of Mathematics Thursday, 8 May 1997 14:00–16:00

Answer as many questions as you can; all carry the same number of marks. Unless otherwise stated, 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}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 \\ 1 & 1 \end{matrix}) , X = (\begin{matrix}1 & −1 \\ 1 & 1 \end{matrix}) , X = (\begin{matrix}1 & 1 \\ 1 & −1 \end{matrix}) .

Which of these 6 matrices X are themselves expressible in the form X = e^{Y}, where Y is a real matrix? (Justify your answers in all cases.)

2. Define a linear group, and a Lie algebra; and define the Lie algebra ŁG of a linear group G, showing that it is indeed a Lie algebra.

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

Are the groups themselves isomorphic?

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)?

4. Define a representation of a Lie algebra; and show how each representation α of a linear group G gives rise to a representation Łα of ŁG.

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, . . ..

5. Show that the only compact connected abelian linear groups are the tori 𝕋^{n}.