Course 424

Group Representations IIIa

Dr Timothy Murphy

Arts Block A2039 Friday, 3 February 1989 15.00–17.00

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 Lie algebra Ł, and a representation of Ł.

Explain how a representation α of a linear group G gives rise to a representation of its Lie algebra ŁG.

Show that if 2 Lie algebras have the same complexification then their representations correspond. Hence or otherwise show that all the representations of sl(2, ℝ) are semisimple.

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

Define also the adjoint representation Ad of the linear group G in ŁG. What is the relation between ad and Ad?

Express the adjoint representation of sl(2, R) in matrix form.

3. Define the Killing form of a Lie algebra.

Compute the Killing form of su(2).

Sketch the proof that if G is a connected linear group, and the Killing form of ŁG is negative-definite, then G must be compact.

4. 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).