 
Summary: Math 7290 Fall 2010
Introduction to Lie Theory P. Achar
Problem Set 1
Due: September 9, 2010
1. Show that GL(n, C) is an algebraic group.
2. Let G = {z C  z = 1}. G is a Lie group under multiplication; it is homeomorphic to a circle. Show
that G is not a complex algebraic group. (It is a "real algebraic group," however.)
3. Let R1 V1 and R2 V2 be root systems. Let V = V1 V2, and let R = R1 R2 V . Let V V
be a subspace, and let T R V be a subset. Show that if T is itself a root system, then the sets
T1 = T R1 V1 and T2 = T R2 V2
are root systems as well, and that T = T1 T2.
4. Classify the irreducible nonreduced root systems. These are related to "super Lie algebras."
(a) Show that for any root system R, if R, then  R.
(b) Recall that the only other possible multiples of that may belong to R are ±1
2 and ±2. Show
that ±1
2 and ±2 cannot both belong to R.
Combining these two steps, we see that there are exactly three kinds of roots:
· A root is reduced if 1
2 / R and 2 / R.
