Course 424

Group Representations III

Dr Timothy Murphy

Joly Theatre Friday, 25 April 2003 15:15–16:45

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, and show that e^{X} is invertible.

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

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

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

2. Define a Lie algebra.

Show that there are just two Lie algebras over ℝ, up to isomorphism; and give two linear groups having these as their Lie algebras.

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.

Do these representations all arise from representations of the group SL(2, ℝ)?

6. What is meant by saying that a topological group G is linearisable? Show that the Lie algebra ℒ G is well-defined in such a case.

Show that the group E(3) (the group of isometries of Euclidean space) is linearisable, and determine its Lie algebra e(3).

7. 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.

Express the group ℂ^{×} in this way.

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