Course 424

Group Representations IIIa

Dr Timothy Murphy

Seminar Room Thursday, 17 June 1999 14:30–16:30

Answer as many questions as you can; all carry the same number of marks. In this exam, ‘Lie algebra’ means Lie algebra over ℝ, and ‘representation’ means finite-dimensional representation over ℂ.

1. Define the exponential e^{X} of a square matrix X.

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

\begin{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 \\ 1 & 1 \end{matrix}) , & X = (\begin{matrix}1 & −1 \\ 1 & 1 \end{matrix}) , & X = (\begin{matrix}1 & 1 \\ 1 & −1 \end{matrix}) . \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 ℒ G of a linear group G, proving that it is indeed a Lie algebra. Define the dimension of a linear group; and determine the dimensions of each of the following groups:

SO(n), SU(n), SL(n, ℝ), Sp(n)?

Continued overleaf 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. 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 there are just two Lie algebras over ℝ of dimension 2 (up to isomorphism); and give two linear groups having these as their Lie algebras.