Math 533, Homework assignment # 1
due on February 18, 2010
Exercise 1: Prove that the exponential map for GL(n, C) is surjective, but not
Exercise 2: Show that GL(n, C) is a connected Lie group.
Exercise 3: Let Fn (C) be the set of all ags in C
Math 533, Homework assignment # 2
due on March 25, 2010
Exercise 1: Let V and W be irreducible representations of a Lie group G. Show
that (V W )G = cfw_0 if V is non-isomorphic to W , and that (V V )G is canonically
isomorphic to C.
Exercise 2: Prove an
Math 533, Homework assignment # 3
due April 20, 2010
Exercise 1: (1) Let g be a reductive Lie algebra. Show that [g, g] is semi-simple.
(2) Let h be an ideal of a Lie algebra g. Show that if both h and g/h are semi-simple
then g is semi-simple.