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. Collections: Mathematics