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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 4
Due: November 18, 2010
1. Prove that every connected abelian unipotent group is isomorphic to a product of copies of Ga. (This
statement is not true in positive characteristic, so you should try to point out where in your proof you
use the fact that C has characteristic 0.)
2. Let G and H be two algebraic groups. Show that Lie(G H) = Lie(G) Lie(H).
3. The groups Gm Gm and Ga Ga have isomorphic Lie algebras: namely, both are 2-dimensional
abelian Lie algebras.
(a) Show that every subalgebra of Lie(Ga Ga) is algebraic.
(b) Classify the algebraic Lie subalgebras of Lie(Gm Gm).
This problem shows that the notion of "algebraic Lie subalgebra" is not, in general, intrinsic to the Lie
algebra, but depends on the algebraic group as well. However, the situation is better for semisimple
Lie algebras.
4. The previous problem shows yields examples where dim a(h) > dim h.
(a) Give examples showing that the difference dim a(h) - dim h can be made arbitrarily large.
(b) Recall that if dim h = 1, then a(h) is a connected abelian group, so we have a(h) = T Gn
a ,
where T is a torus, and U is a connected abelian unipotent group. Can every connected abelian


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


Collections: Mathematics