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Math 7290 Fall 2010 Introduction to Lie Theory P. Achar

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.


Source: Achar, Pramod - Department of Mathematics, Louisiana State University


Collections: Mathematics