Lie groups and Lie algebras: Questions 2

1. Derivative of A(t)^-1

(Depends only on material from Week 1.)

Let A(t) be a family of matrices depending smoothly on a parameter t.

1. Is it true or false that d by d t of A to the n equals n A to the n minus 1 times d A by d t for all positive n? Give a proof or a counterexample.

2. Suppose that A(t) is invertible for all t. By differentiating the condition A(t) times A(t) inverse equals the identity, find a formula for d by d t of A(t) inverse.

2. More about tangent spaces

(Depends on the video about the Lie algebra as the tangent space).

Suppose that gamma from R to big G is a path in a matrix group big G.

1. Fix s in R and consider the path delta of t given by (gamma of s) inverse times gamma of (s + t). Find delta at 0 and delta dot at 0.

2. Hence or otherwise, show that gamma of s inverse times d gamma by d s is an element of the Lie algebra little g.

3. The tangent space to big G at some point g in big G is defined to be the set of all tangent vectors to paths gamma of t in big G such that gamma of zero equals g. Deduce from the previous part of the question that the tangent space to big G at g equals g times little g.

3. Topological closure

(Depends on the video about matrix groups.)

Show that the topological closure of a subgroup G inside G L n R is a subgroup of G L n R. [Hint: You may use the fact that the topological closure of G is the set of all matrices in G L n R arising as limits of sequences of matrices M_k in G.]

4. Heisenberg group

(Depends on the Lie algebra of a matrix group and the notion of commutator.)

The Heisenberg group H_3 is the group of 3-by-3 matrices of the form 1, a, c; 0, 1, b; 0, 0, 1.

1. What is its Lie algebra?

2. Verify that the exponential map is bijective for this group.

3. Find a basis for the Lie algebra such that each basis element has at most one nonvanishing matrix entry, and compute the commutators between your chosen basis elements.

5. Alternative proof that little g is a Lie algebra

(Depends on the videos about little g as a tangent space and as a Lie algebra.)

Let G be a matrix group with Lie algebra little g. We will figure out an alternative proof that little g is preserved by Lie bracket.

Suppose that X and Y are in little g. Let Z be the matrix exp of s X times Y times exp of minus s X. By considering exp t Z, show that Z is in little g for all s in R. Why does this imply X bracket Y is in little g?

(Hint: If gamma of s is a path in a vector space then its tangent vector also belongs to this vector space. You may assume that little g is a vector space.)

6. A subrepresentation

(Builds on the video about complete reducibility).

Recall that a representation is a homomorphism R from G to G L n R, and a subrepresentation is a subspace V inside R n such v in V implies R of g v is in V for all g in G.

Let G be the group of real numbers with addition. Consider the homomorphism R from G to G L 2 R, R of t equals 1, t, 0, 1. Show that the x-axis is a subrepresentation (i.e. that R(t) preserves the x-axis for all real numbers t). Are there any other subrepresentations? (Either exhibit one explicitly or prove there are none). Is R completely reducible?

7. Symplectic group

(Depends on the idea of a matrix group and its Lie algebra).

Let Omega be the 2 n-by-2 n matrix given by n-by-n blocks Omega equals 0, I, minus I, 0 Define S p 2 n, R to be the set of matrices M in G L 2 n, R such that M transpose times Omega times M equals Omega.

1. Check that S p 2 n, R is a group. Is it topologically closed?

2. Find the Lie algebra little s p 2 n, R.

3. Show that S p 2 n, R is an unbounded group (i.e. you can find a sequence of matrices M_k in S p 2 n, R such that some of the matrix entries of M_k go to infinity as k goes to infinity.

4. Show directly that if X and Y are in little s p 2 n, R then X bracket Y is in s p 2 n, R. Is it true that X times Y is also in little s p 2 n, R for all X and Y in little s p 2 n, R?

8. Lorentz group

(Builds on the example homomorphism from SU(2) to SO(3).)

Let c in R be a positive constant (the speed of light!). Define eta to be the 4-by-4 matrix minus c squared, 1, 1, 1 down the diagonal and zeros elsewhere and define O(1,3) to be the group of matrices M in G L 4 R such that M transpose eta M equals eta This is called the Lorentz group: it consists of transformations of 4-dimensional spacetime preserving the "spacetime interval" minus c squared t squared plus x squared plus y squared plus z squared.

1. Let zeta be a real number. Calculate exp of 0, zeta over c, 0, 0; c zeta, 0, 0, 0; 0, 0, 0, 0; 0, 0, 0, 0 and show it lives in O(1,3). This is called an (inverse) Lorentz boost, and it's the simplest Lorentz transformation which mixes up space with time.

2. Let's encode vectors v = (t, x, y, z) as 2-by-2 matrices S_v equals c t minus x, y plus i z, y minus i z, c t plus x. Calculate det of S_v

3. Show that S_v is Hermitian, that is S_v dagger equals S_v. Moreover, show that any Hermitian matrix is of the form S_v for some v in R 4.

4. Given a matrix A in S L 2 C, check that A S_v A dagger is Hermitian (note that this implies A S_v A dagger equals S v prime for some v prime by the previous part of the question).

5. Define F from S L 2 C to O(1, 3) implicitly by S subscript F of A applied to v equals A times S_v times A dagger. Show that this is a homomorphism.

In fact, this homomorphism is 2-to-1 onto the identity component of the Lorentz group, so S L 2 C is basically the spin group associated to O(1,3).

9. Discontinuous homomorphism

(Depends on the video about smooth homomorphisms.)

In this course, we're focusing on smooth homomorphisms, but it's quite possible to have homomorphisms which are not smooth (even discontinuous). In this question, we'll "construct" one.

1. Explain the sentence "R is a vector space over Q". Give me an example of two elements of R which are linearly independent over Q.

2. For the rest of the question, we will assume we have picked a basis B for this vector space. Note that this vector space is so awful that nobody has ever written down (or will ever write down) a basis explicitly. Can you explain why specifying a basis would be such a difficult problem?

3. For each b in B, pick a real number lambda_b. Define F from R to R by F of sum c_b times b, equals sum of lambda_b times c_b times b, (here, the notation means that sum c_b times b is the unique vector whose b-component is c_b). If we equip R with addition, show that F is a homomorphism.

4. Suppose not all the lambda_b are equal. Why is F not smooth?

10. Neighbourhood of the identity generates

(Technically no prerequisites, but requires you to know the Heine-Borel theorem from analysis. This is a hint about how to solve the problem!)

Let G be a matrix group and let S inside G be an open neighbourhood of the identity matrix.

1. By considering the set S inverse, the set of inverses of elements in S show that there is a smaller open neighbourhood S_1 of the identity such that g\in S_1 implies g^{-1}\in S_1. Henceforth, we will drop the subscript 1 and just assume that S always had this property.

2. Let gamma from the interval 0,1 to G be a continuous path in G with gamma of 0 = the identity. Prove that g equals s_1 times s_2 times dot dot dot times s_N for some sequence of elements s_i in S. You may use the following diagram as inspiration for coming up with the proof, provided you explain what it's trying to depict and how it is constructed.

   

This tells us that any neighbourhood of the identity generates the path component of the identity in G.

11. More about the Heisenberg group

(Depends on the earlier question about this group and the video about smooth homomorphisms.)

Consider the subgroup of 3-by-3 matrices 1, 0, c; 0, 1, 0; 0, 0, 1 with c a real number, inside the Heisenberg group H_3.

1. Show that elements of N_3 commute with everything in H_3.

2. Show that elements of N_3 can be written in the form ghg^{-1}h^{-1} for some g,h\in H_3.

3. Deduce that if R from H_3 to G L n C is a homomorphism then \det(R(g))=1 for all g\in N_3.

We will use these results in a later exercise to construct a (Lie) group which is not a matrix group.

12. Fourier theory

(Depends on the video about smooth homomorphisms).

Let V be the (infinite-dimensional) vector space of complex-valued 2 pi-periodic functions f of theta (such that f of theta plus 2 pi equals f of theta). We will define a homomorphism R from U(1) to G L V, where G L V now denotes the space of invertible linear maps from V to V. Define R of e to the i phi to be the linear map which takes a function f of theta to f of (theta plus phi).

1. Check that R of e to the i phi is linear.

2. Check that R is a homomorphism.

3. Fix an integer n. Suppose that f is an eigenvector of R of e to the i phi with eigenvalue e to the i n phi for all phi. I claim this completely determines f up to an overall constant factor: what is f?

We will later see that these are the only possible eigenvalues.