**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 R^{3} with binary composition
law given by the cross-product, is a Lie Algebra. Show that this
Lie algebra is isomorphic to the Lie algebra
sl_{2}(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.