MATH 743 Linear Algebraic Groups (Fall 2002)

# MATH 743: Linear Algebraic Groups

### Homework Questions

#### Set II

(There are 10 questions.)

Let G=GL4(K) and D=D4(K) be the subgroup of G consisting of diagonal matrices. If possible, write 5 tori in D with non-isomorphic centralizers CG(T).

Section 21: Questions 2, 4, 6, 9

Section 22: Questions 1, 6

Section 23: Questions 3, 4 (the first part only), 8

### Set I

(The homework is due on 31 October 2002 Thursday 12:40.)

K is an algebraically closed field.

1. Find an example of a subset X of Kn such that X is different from V(I(X)).
2. Show that a point derivation is uniquely determined by its action on K[G], where G is an algebraic group.
3. Find the Lie algebra of SLn(K). Justify your claim.
4. Let X, Y, Z be affine varieties, P a point in X, f a morphism from Y to Z and g a morphism from X to Y. Show that d(f*g)P= dfg(P)*dgP. [Here * stands for the composition of maps.]
5. If H is an abelian subgroup of an algebraic group G then show that the closure of H is also an abelian subgroup of G.
6. Exercise 1.11.
7. Exercise 7.4.
8. Exercise 7.11.
9. Exercise 8.1.
10. Exercise 10.2.
Good luck!