Homework Set #1

1. Show that C[x] is a Lie algebra with respect to the bracket [P(x) Q(x)] = P'(x)Q(x) - Q'(x)P(x).
2. Let L be a Lie algebra and x,y∈L. Prove that [[[x y] x] y] = [[[x y] y] x].
3. Prove that every two-dimensional Lie algebra has a one-dimensional ideal.
4. Prove that for n≥2, the Lie algebra of complex trace-0 n×n matrices with respect to the commutator is perfect.
5. Prove that C³ under cross product, complex trace-0 2×2 matrices under commutator, and complex skew-symmetric 3×3 matrices under commutator are all isomorphic to one another.