189-571B: Higher Algebra II

Assignment 8

To be presented: Wednesday, March 14.

The problem marked with a (*) is for everyone to do and to hand in on the day of the problem session. I will return this problem to you on Friday of the same week, so late problems will not be accepted!

If you are stuck on your problem, come see me during my Monday office hours (or make an appointment) to get a hint...

1. (*) Show that the vector space R³ with binary composition law given by the cross-product, is a Lie Algebra. Show that this Lie algebra is isomorphic to the Lie algebra sl₂(R) seen in class.

Oops! There is a mistake in the above question. What I REALLY should have said is: show that these two algebras are NOT isomorphic over R, but their tensor products over C become isomorphic as Lie algebras over the complex numbers. (The tensor product of a Lie algebra over F by a field extension E just the usual vector space tensor product, with the algebra operation defined in the same way as we did for associative algebras...)

2. (Matt Greenberg) a) Classify the two-dimensional Lie Algebras over R, and show that there are exactly two such algebras, up to isomorphism.

b) For each such algebra A, exhibit a Lie group over R whose tangent space at the identity is isomporphic to A.

c) Do the same for the three dimensional Lie algebras over R.

3. (Kristina Loeschner) Let L be a Lie algebra, and let ad: L --> end(L) be its adjoint representation.

a) Show that ad(x) is a derivation on L.

b) A derivation on L is called inner if it is of the form ad(x) for some x in L. Show that the set of inner derivations is an ideal of the Lie algebra Der(L) of all derivations on L.