Math 319 Paper Assignment Lie Groups

Your paper will be on one of the topics in the following list. (If there is something not on the list that you would particularly like to explore, come talk to me about it.) You may use any books or articles as resources in preparing your paper, but all sources should be acknowledged. You may discuss your work with other students or with me (in fact, this is encouraged!), but the writing should be your own.

Your paper should be written in narrative form, in complete sentences and should be 4-6 typed pages in length. The final version is due at 5pm on December 6. Imagine your audience is another student with the same mathematical background as yours and that you are explaining your ideas to her. Provide the definitions of any terms not already defined in class and illustrate new terms with examples. You may cite previous results (from your reading or from class) and use them without proof, as needed. Length considerations will help determine which results you prove in full and which you merely cite. Include at least one proof. Be sure to verify that the necessary hypotheses are satisfied when you apply a theorem.

Whenever possible, relate your topic to the course material. Explain the meaning of any result you prove, and provide enough detail in your argument so that the reasoning would be clear to someone encountering your proof for the first time. Illustrate the results with one or more examples to make the meaning clearer.

You should review your intended topic with me by October 28 so that I can be sure you have the appropriate background to handle it effectively. As a general guideline, if you haven't taken Math 311, Abstract Algebra, choose from among topics 1 - 11. No later than November 22, you should give me an outline indicating which statements you will prove and what examples you will use, so I can be sure you aren't attempting too much or too little. The completed paper is due by 5pm on December 6.

Possible topics (sorry for the TeX notation for mathematical symbols):

1. Discuss world lines and the partition of spacetime. Show that the rotations of the first kind map the ``future set" one-to-one and onto itself. Reference: Callahan, pp. 1-9 and pp. 43-59.

2. Elaborate the analogy between the circular and hyperbolic trigonometric functions by parametrization by area and by paramaterization by arc length. Reference: Callahan, pp. 43--45, 61-62, and 140, and Taylor and Wheeler, p. 59.

3. Prove the Cartan decomposition for SO(3, R). Reference: Howe and Barker, Chapter XII, p. 42 ff. (See Sattinger and Weaver, pp. 10-12 for a different treatment; also, compare P.S. 5, # 6, 7.)

4. Show that {\cal L}^{++}(4, R) is generrated by spatial rotations and Lorentz boosts. Reference: Howe and Barker, Chapter XV, pp. 52-55.

5. Give an exposition of hyperbolic geometry, including at least some of the exercises.

Reference: Howe and Barker, Chapter XIV, pp. 11-16.

6. Do the exercises illustrating Howe's claim that many of the important classical

differential equations are related to Lie theory. Reference: Howe, p. 622, Exercises A and B.

7. Prove that the Moebius transformations coming from SL(2,R) are isometries of the Poincare half plane. More specifically, show that for the matrix

$$

\left[ \begin{array}{cc}

a & b \\

c & d

\end{array} \right] \; \in \; SL(2, \bbr)

$$

the coresponding Moebius transformation m(z) = w = \frac{az + b}{cz + d}

takes z=x+iy with Im z = y>0 to w = x' + y' i with Im w = y' > 0. Also the metric on the upper half plane is preserved by m$. That is,

\frac{dw \, d\ow}{(\mbox{Im} \, w)^2}\, = \, \frac{dz \, d\oz}{(\mbox{Im} \, z)^2}

where dw = (dw/dz)dz = [1/(cz + d)^2] dz and d \ow = [1/(c \oz + d)^2] d \oz. Reference: Sattinger and Weaver, pp. 8-10. (Also see Mumford, et al, pp. 69-77 on Moebius transformations coming from GL(2, R).)

8. The group SU(2, C) is the covering group of the Euclidean group SO(3,R). Show that there is a group homomorphism mapping SU(2, C) onto SO(3, R). (Notice that the transformation m in the previous item can be thought of as a function C \cup \{ \infty\} \rightarrow C \cup \{ \infty \} because m takes -d/c to \infty and takes \infty to a/c.) Reference: Sattinger and Weaver, chapter 4, pp. 10-16.

9. The group SL(2, C) is the covering group of the Lorentz group {\cal L}^{++}(4, R). Show that there is a group homomorphism mapping SL(2, C) to {\cal L}(4, R). Specifically (using notation different from the reference), show that the following statements are true.

a. The vectors \bv=(t, x, y, z) of space time can be represented by matrices

$$

V = \left[ \begin{array}{cc}

t+z & x-iy \\

x+iy & t- \oz

\end{array} \right],

$$

with the Lorentz product \bv \cdot \bv = \det(V).

b. The coordinates of \bv can be recovered from the entries of V by t = (1/2) tr(V), x= (1/2) tr(V \sigma_1), y = (1/2) tr(V \sigma_2), z= (1/2) tr(V \sigma_3), where the \sigma_j are the Pauli spin matrices.

c. For A \in SL(2, C), define T_A: R^4 \rightarrow R^4 by T_A(V) = AV \oA^t. Show T_A \in {\cal L}(4, R). (In fact, T_A is a rotation of the first kind, so the map \Phi(A) = T_A is actually from $SL(2, C) to {\cal L}^{++}(4, R).)

d. Show that \Phi(A) = T_A defines a group homomorphism. Reference: Sattinger and Weaver, pp. 16-18.

10. Show that the set S^3 of quaternions a + bi + cj + dk with a^2 + b^2 + c^2 +d^2 = 1 is a group under multiplication and is isomorphic to SU(2, C). Reference: Artin, p.306, problem 1. Also see Burn, p. 179 and Zulli, pp. 222-223.

11. Describe the intrinsic distance on the 3-sphere S^3 and show that conjugation by elements of the group S^3 is an isometry preserving this distance. Reference: Zulli, pp. 223-225.

12. Describe the relationship between longitudes of the 3-sphere and conjugacy classes of the group S^3. Reference: Zulli, pp. 225-227, or Artin, pp. 273-276.

13. Prove that the group of inner automorphisms of the quaternions is isomorphic to the (Euclidean) group SO(3, R). Reference: Burn, chapter 19, pp. 179-183. (Note that codomain means the same as range, and bijection means the same as one-to-one and onto function.)

14. Prove that if G is a connected Lie group then G is abelian if and only if its Lie algebra L(G) is abelian. (A Lie algebra L is abelian if [X,Y]=0 for all X,Y \in L.) Reference: Ise and Takeuchi, p. 34ff. (You can use without proof the prior results needed for this argument.)

15. Prove that the normalizer of a Lie subalgebra is a Lie subalgebra. Also prove that the centralizer of a Lie subalgebra is a Lie subalgebra. Reference: Humphreys, pp. 6-7.

16. Prove that the derivations $Der(L)$ of a Lie algebra $L$ form a Lie algebra and that the inner derivations form an ideal of $Der(L)$. Reference: Humphreys, pp. 4, 6.

17. Suppose that F is a finite field containing q elements. (If q is a prime, you may think of F as the set {0,1,2,..., q-1} with addition and multiplication defined modulo q --- sometimes called clock arithmetic.) Show that GL(2,q), the group of all invertible 2 by 2 matrices with entries in F, contains (q^2-1)(q^2-q) matrices. Also show that SL(2,q), the subgroup of GL(2,q) consisting of matrices of determinant one, contains (q^2-1)(q^2-q)/(q-1) = (q^2-1)q elements. Reference: Gorenstein, p.40.

18. (Refer to the previous item for notation.) Show that SL(2,q) contains cyclic

subgroups of orders q-1 and q+1. Under what circumstances is there a cyclic subgroup of order q? Reference: Gorenstein, p. 42.

19. Investigate the Lie algebra isomorphisms among the classical Lie algebras of small rank. In particular, working over R (or over an arbitrary field), show that the classical Lie algebras of type A_1, B_1 and C_1 are all isomorphic. Show that the Lie algebra of type D_1 is 1-

dimensional. Show that the Lie algebras of type B_2 and C_2 are isomorphic, as are the Lie algebras of type D_3 and A_3. What can you say about the Lie algebra of type D_2? Reference: Humphreys, pp. 1-6 (see exercise 10 on p.6).

20. Prove that for a connected Lie group G, if U is a neighborhood of the identity, then every element of G is a finite product of elements of U. Reference: Hausner and Schwartz, Lemma 2, p. 37.

21. This is an alternate approach to some of the ideas we are treating in

class. For a matrix group G \subseteq GL(n, R), define \cal G = { A \in M(n, R) : exp(tA) \in G for all t \in R } Prove that

a) \cal G is a Lie algebra, and

b) exp: \cal G \rightarrow G maps a neighborhood of 0 in \cal G one-to-one and onto a neighborhood of the identity in G. Reference: Howe, Theorem 17 and Lemma 18, pp. 616-618.