Course 424

Group Representations III

Dr Timothy Murphy

Arts Block A2039 Friday, 20 January 1989 15.45–17.45

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

Which of these 4 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 SO(3) and SL(2, ℝ), and show that they are not 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 SU(2); and show that this Lie algebra su(2) has just 1 simple representation of each dimension 1, 2, 3, . . ..

4. 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