Exercise List #4

Classical Field Theory

November 4, 2009

1. Check the properties of algebraical (ie. under group operations) and topological (ie.

with respect to the Euclidean norm

P |a

mn

|

2

) closure of the following subgroups of

GL(n): SL(n), U (n), SU (n), O(n), SO(n), SO(p, q).

2. Prove the following (Jacobi) identity for matrix algebras. Express it in terms of

structure constants.

[[A, B], C] = [A, [B, C]] − [B, [A, C]]

3. Prove that, for any matrix X, the one-parameter subgroup of GL(n)

{A(t) = exp tX : t ∈ R}

is indeed a group.

4. Check that a Lie algebra g of a Lie group G is indeed an algebra, ie. for any X, Y ∈ g

also X + Y, [X, Y ] ∈ g.

5. Find Lie algebras for Lie groups mentioned in Ex. 1.