Course 424

Group Representations III

Dr Timothy Murphy

Joly Theatre  Friday, 4 May 2001  14:00–15:30

Attempt 5 questions. (If you attempt more, only the best 5 will be counted.) All questions 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}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}0 & 1 \\ 0 & 0 \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, showing that it is indeed a Lie algebra.

Show that a homomorphims of linear groups f : G → H gives rise to a Lie algebra homomorphism ℒ f : ℒ G → ℒ H

If f is surjective, does it necessarily follow that ℒ f is surjective? If f is injective, does it necessarily follow that ℒ f is injective? (Give reasons.)

Continued overleaf 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), {Sp(n), E(n)?

(E(n) is the isometry group of n-dimensional Euclidean space.)

4. 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.

5. 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.

6. 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′.