Course 424

Group Representations III

Dr Timothy Murphy

22 September 1987 Exam Hall 14:00–16:00

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 a representation α of a Lie algebra, and explain carefully how a representation of a linear group G gives rise to a representation of its Lie algebra ℒ G.

What is meant by saying that α is (a) simple (b) semisimple?

Determine the simple representations of sl(2, ℝ).

Sketch the proof that all representations of sl(2, ℝ) are semisimple.

2. Define the adjoint representation ad of a Lie algebra ℒ (verifying that it is indeed a representation).

Compute the adjoint representation of so(3) in matrix form.

Define a covering f : G → H, where G and H are connected linear groups. By using the adjoint representation Ad of the group SU(2), or otherwise, establish that there is a covering f : SU(2) → SO(3). What is ker f?

3. Define the Killing form K(X) of a Lie algebra.

Compute the Killing forms of su(2) and sl(2, ℝ). Show that the Killing form of a compact linear group G (or rather, of ℒ G) is negative-definite or negative-indefinite. Sketch the proof of the converse result, that if the Lie algebra ℒ G of the connected linear group G is negative-definite then G is compact.

4. Show that any set of commuting matrices has a common eigenvector. Sketch the representation theory of su(3), defining carefully the terms weight and weight vector. Determine the weights of the adjoint representation of su(3).