 
Summary: Math 7290 Fall 2010
Introduction to Lie Theory P. Achar
Problem Set 3
Due: October 19, 2010
1. Prove that the set of diagonal matrices in sln is a Cartan subalgebra. (We used this example in class
many times, but we never actually proved this statement.)
2. Let X be an affine variety in CN
. Prove that X has only finitely many connected components. (Hint:
Show that for any component X1 X, there is a polynomial f C[x1, . . . , xN ] such that fX X1 = 0
but fX1
= 0. Then use the fact that C[x1, . . . , xN ] is a noetherian ring.)
3. Let X CN
be a variety defined by equations
f1(x1, . . . , xN ) = f2(x1, . . . , xN ) = · · · = fk(x1, . . . , xN ) = 0.
Assume that this set of equations is reduced, meaning that every polynomial p C[x1, . . . , xN ] that
vanishes on all points of X is in the ideal generated by f1, . . . , fk. Consider a point a = (a1, . . . , aN )
X. Show that the tangent space TzX can be identified with the following subspace of CN
:
(b1, . . . , bN ) for all j,
N
