Announcements

We are meeting in Math 0106 starting Wednesday, August 31.

Office hours 1:30-3, or by appointment, in 4412.

The last problem and discussion session with Vincent will take place during normal class time Monday 12/12 12-12:50 pm. You are welcome to attend my office hours 1:30-3 pm for further discussion.

Problem Sets

• HW 1, due 9/14 (FINAL)
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  1. Let G be a bounded matrix group, so in particular, G is closed in GL_n(C). Show that G is closed in C^n×n, thus G is compact.
  2. Show that an elliptic matrix in GL_n(R)---one which is diagonalizable over C with eigenvalues all of modulus one---is conjugate in GL_n (R) to an orthogonal matrix. (Hint: The nonreal eigenvalues come in conjugate pairs, and so do their eigenvectors in Cⁿ.)
  3. (a) Show that SL_n(R) deformation retracts to SO(n) (Hint: Gram-Schmidt orthonormalization) .
     (b) Show that O(p,q) deformation retracts to O(p) × O(q) (Hint: Consider the space Gr⁺(p,p+q) of p-dimensional positive definite subspaces of R^p,q. Find a deformation retract of this space to the positive definite subspace span{e₁, . . ., e_p}.)
  4. Let Γ be a discrete subgroup of a Lie group G. That means that every subset of Γ is open in the subspace topology. Show that Γ is closed in G. (This fact is true in greater generality for topological groups, but some assumptions on the topology are required.)
  5. In class on 9/7, we showed that given a covering G^~ → G of a connected Lie group G, we can lift the multiplication map m and the inversion map ι of G to continuous maps on G^~. Verify the following two properties of these lifts, which are also denoted m and ι, respectively:
     • For any g in G^~, m(ι(g),g)= I^~.
     • For any g,h, and k in G^~, m(m(g,h),k) = m(g,m(h,k)).
  6. Hall Exer. 1.6.3: Show that the symplectic group Sp₁(R) < GL₂(R) is isomorphic to SL₂(R). Show that Sp₁(C) is isomorphic to SL₂(C).
  7. Hall Exer. 1.6.5: Show that if α and β are arbitrary complex numbers, satisfying |α|² + |β|² = 1, then the matrix $$A = \left( \begin{array}{cc} \alpha & - \overline{\beta} \\ \beta & \overline{\alpha} \end{array} \right) $$ is in SU(2). Show that every A in SU(2) can be expressed in this form for a unique pair α, β satisfying |α|² + |β|² = 1.
  8. Hall Exer. 1.6.17: Suppose G < GL_m(C) and H < GL_n(C) are matrix Lie groups and that Φ : G → H is a Lie group homomorphism. Then the image of G under Φ is a subgroup of H and thus of GL_n(C). Is the image of G under Φ necessarily a matrix Lie group? Prove or give a counter-example.

• HW 2, due 9/28 (FINAL)
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  1. Hall Exer. 2.6.6: Show that every 2 × 2 matrix X with trace(X)=0 satisfies X² = - (det X) I. If X is 2 × 2 with trace zero, show by direct calculation using the power series for the exponential that $$ e^X = \cos \left( \sqrt{\mbox{det} X} \right) I + \frac{\sin \sqrt{\mbox{det} X}}{\sqrt{\mbox{det} X}} X,$$ where \( \sqrt{\mbox{det}X} \) is either of the two (possbily complex) square roots of det X. Use this to give an alternative computation of the exponential of $$ X = \left( \begin{array}{cc} 0 & - a \\ a & 0 \end{array} \right)$$
  2. Hall Exer. 2.6.7: Use the previous exercise to compute the exponential of $$ X = \left( \begin{array}{cc} 4 & 3 \\ -1 & 2 \end{array} \right)$$ Hint: Reduce the calculation to the trace-zero case.
  3. Hall Exer. 2.6.9: A matrix A is unipotent if A-I is nilpotent. Note that log A is defined whenever A is unipotent, because the series terminates.
     • Show that if A is unipotent, then log A is nilpotent.
     • Show that if X is nilpotent, then e^X is unipotent.
     • Show that if A is unipotent, then exp(log A) = A, and that if X is nilpotent, then log(exp X) = X. Hint: Let A(t) = I + t(A-I). Show that exp(log(A(t))) depends polynomially on t and that exp(log(A(t))) = A(t) for all sufficiently small t.
  4. Show that the following matrix is not in the image of the exponential map of GL₂(R): $$ \left( \begin{array}{cc} -1 & 1 \\ 0 & -1 \end{array} \right) $$
  5. Let Φ: G → H be a homomorphism of Lie groups, with H connected, and suppose that φ = D_IΦ is an isomorphism. Show that Φ is a covering map of Lie groups. (You should use the proposition on coverings of Lie groups we proved in class.)
  6. Verify that the center of SL₂(R) is ± Id₂, and similarly for Z(SU(2)).
  7. Hall Exer. 3.9.2: Classify up to isomorphism all one-dimensional and two-dimensional real Lie algebras. There is one isomorphism class of one-dimensional algebras and two isomorphism classes of two-dimensional algebras.
  8. Hall Exer. 3.9.9: Write out explcitly the general form of a 4 × 4 real matrix in \( \mathfrak{o}(1,3) \).
  9. Hall Exer. 3.9.16: Suppose G is a connected Lie group (where we define the Lie algebra in the "standard" way, in which the underlying vector space is T_IG, not as in Hall) which is also commutative. Show that the exponential map is onto from the Lie algebra of G to G.

• HW 3, due 10/12 (FINAL)
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  1. (a) Define the matrix group O_n(C) to comprise the elements of GL_n(C) which preserve the quadratic form on Cⁿ $$ Q({\bf v}) = v_1^2 + \cdots + v_n^2 $$ Find the Lie algebra o_n(C).
     (b) Show that o(p,q)_C ≅ o_n(C), where n=p+q.
  2. Given a real Lie algebra g or a homomorphism of real Lie algebras φ, denote their complexifications g_C and φ_C, respectively; similarly, denote the realification of a complex Lie algebra h or a homomorphism ψ of complex Lie algebras by h_R and ψ_R, respectively. In class on 9/28 we defined a function F : Hom_R(g,h_R) → Hom_C(g_C,h).
     • Find a function Hom_C(g_C,h) → Hom_R(g,h_R) that is inverse to F.
     • Show that F is natural in g. That is, given a homomorphism α: g → g', the complexification α_C satisfies α^*_C • F = F • α^*.
     • Show that F is natural in h. That is, give a homomorpism β: h → h', the realification β_R satisfies β • F = F • β_R.
     Congratulations, you have shown that complexification and realification are a pair of adjoint functors!
  3. Consider the following basis for the Lie algebra sl₂(R): $$ H = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) \qquad E = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right) \qquad F = \left( \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right) $$
     • Find the left-invariant vector fields H^†, E^†, and F^† on SL₂(R) corresponding to H, E, and F, as functions of $$ g = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right)$$
       
     • Compute the flows along H^†, E^†, and F^†, as a function of time and of initial point g ∈ SL₂(R).
     • Compute the vector field brackets [H^†,E^†], [H^†, F^†] and [E^†,F^†].
       
     • Compute the algebraic brackets in the Lie algebra [H,E], [H,F], [E,F].
       

  4. Hall Exer. 5.11.6: Give an example of matrices X and Y in sl₂(C) such that [X,Y] = 2 π iY but such that there does not exist any Z in sl₂(C) with e^X e^Y = e^Z .
  5. Hall Exer. 5.11.12: Show that every connected, injectively immersed, Lie subgroup of SU(2) is closed. Show that this is not the case for SU(3).

• HW 4, due 10/26 (FINAL)
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  1. Prove: For G a connected Lie group, and (ρ₁,V₁), (ρ₂, V₂) finite-dimensional representations of G with corresponding Lie algebra representations (φ₁, V₁), (φ₂, V₂), respectively, the Lie group representations (ρ₁,V₁) and (ρ₂, V₂) are isomorphic if and only if the representations (φ₁, V₁) and (φ₂, V₂) are isomorphic.
  2. Show that a representation (ρ,V) of a Lie group G is self-dual---that is, (ρ^*,V^*) ≅ (ρ,V)---if and only if there is a nondegenerate quadratic form Q on V for which ρ(G) ≤ O(Q).
  3. Give a two-dimensional representation of \( \mathfrak{g} = \mathfrak{o}(3, {\bf C}) \).
  4. Hall Exer. 4.9.5: Consider the standard representation of the Heisenberg group, acting on C³. Determine all invariant subspaces of C³. Is this representation completely reducible?
  5. Consider the representation \( \wedge^2 {\bf R}^{2,1*} \otimes \mathfrak{o}(2,1) \) of \( \mathfrak{o}(2,1) \cong \mathfrak{s}\mathfrak{l}_2({\bf R}) \).
     • Construct an isomorphism to the representation on End(R³) by bracket.
     • Decompose this latter representation into irreducible components.
     • (I decided to cancel this last part.)
  6. Show that the even-dimensional irreducible representations π_2m-1 of sl₂(R) preserve a nondegenerate skew-symmetric bilinear form. Show that the odd-dimensional irreps π_2m preserve an inner product of split signature (m,m+1).
  7. Let \( \mathfrak{g} \) be a nilpotent Lie algebra.
     • Show that \( \mathfrak{g} \) admits a filtration by ideals $$ \mathfrak{g} = \mathfrak{g}_0 \rhd \mathfrak{g}_1 \rhd \cdots \rhd \mathfrak{g}_{n-1} \rhd \mathfrak{g}_n = 0$$ Such that \( \mathfrak{g}_i / \mathfrak{g}_{i+1} \cong {\bf R} \) for all i = 0,. . ., n-1.
     • Given H and G any simply connected Lie groups, let \( \varphi : \mathfrak{g} \rightarrow \mbox{Der} \mathfrak{h} \) be a homomorphism. Then there are a semidirect product Lie algebra \( \mathfrak{g} \ltimes_\varphi \mathfrak{h} \); a homomorphism \( \Phi: G \rightarrow \mbox{Aut} H \) with \( D_I \Phi = \varphi \); and a semidirect product Lie group \( G \ltimes_\Phi H \) with Lie algebra \( \mathfrak{g} \ltimes_\varphi \mathfrak{h} \), in which G and H are closed subgroups. (I'm just telling you; you don't have to prove this.)
       
       Now show that for \( \mathfrak{g} \) nilpotent as before, and \( \mathfrak{h} \lhd \mathfrak{g} \) codimension one, \( \mathfrak{g} \cong {\bf R} \ltimes \mathfrak{h} \).
     • Show that for any nilpotent Lie algebra \( \mathfrak{g} \), there is a simply connected nilpotent Lie group G with this Lie algebra.
     • Next show that the normal subgroups G_(i) in the lower central series for a simply connected nilpotent Lie group G (called the upper central series in Hall, but not in most other references) are closed.
     • Finally, show that the exponential map is onto for G a simply connected nilpotent Lie group (Hint 1: Use the BCH formula; Hint 2: If H is a closed normal subgroup of a Lie group G, then G/H is a Lie group.).

• HW 5, due 11/9 (FINAL)
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  1. Let \( \mathfrak{h} \lhd \mathfrak{g} \) be a Lie algebra ideal. Show that the Killing form \( K_{\mathfrak{h}} \) equals the restriction \( \left. K_{\mathfrak{g}} \right|_{\mathfrak{h}} \).

  2. Show that a semisimple ideal in any Lie algebra is a direct summand.
  3. Consider the group $$ \mbox{SOL} \cong {\bf R}^\times \ltimes_\varphi {\bf R}^2 \qquad \varphi(\lambda)(x,y) = (\lambda x, \lambda^{-1} y)$$
     • Compute the Lie algebra of SOL.
     • Compute ad for \( \mathfrak{sol} \).
     • Compute the Killing form for \( \mathfrak{sol} \).
     • Compute the Killing form for \( \mathfrak{o}(3) \).

  4. Hall Exer. 7.8.4, first part: Using the root space decomposition in Sect. 7.7.2, show that the Lie algebra \( \mathfrak{o}(4,{\bf C}) \) is isomorphic to the Lie algebra direct sum \( \mathfrak{sl}(2,{\bf C}) \oplus \mathfrak{sl}(2,{\bf C}) \).
  5. Hall Exer. 7.8.8

• HW 6, due 11/30 (FINAL)
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  1. Show that if \( \mathfrak{h} \) is a maximal abelian, simultaneously ad-diagonalizable subalgebra of a semisimple Lie algebra \( \mathfrak{g} \), then it is a Cartan subalgebra---that is, \( \mathfrak{h} \) is nilpotent and equals its normalizer.
  2. Let ω be the standard symplectic form on R⁴, and view it as \( e_1^* \wedge e_3^* + e_2^* \wedge e_4^* \in V = \wedge^2 ({\bf R}^4)^* \). Note that \( \omega \wedge \omega = -2 \cdot \mbox{vol} \) where vol is the standard volume form on R⁴.
     • Define an inner product B on V by $$ \alpha \wedge \beta = B(\alpha,\beta) \cdot vol$$ Show that B is symmetric.
     • Restrict B to \( W = \omega^\perp \subset V \), and show that it has type (3,2).
     • We have defined a homomorphism Sp(2,R) → O(3,2). Show that the kernel is discrete and that the image has nontrivial connected component of the identity (It suffices to exhibit a nontrivial 1-parameter subgroup in the image.)
     • Compare dimensions of these two groups, and conclude that they are isogeneous, so \( \mathfrak{sp}(2,{\bf R}) \cong \mathfrak{o}(3,2) \), and thus \( \mathfrak{sp}(2,{\bf C}) \cong \mathfrak{o}(5,{\bf C}) \).
     • Alternatively, examine the root systems for the complex Lie algebras \( \mathfrak{sp}(2,{\bf C}) \) and \( \mathfrak{o}(5,{\bf C}) \) given in subsections 7.7.4 and 7.7.3 of Hall, respectively. What is the bijection between these roots that gives an isometry of the root systems?
  3. Let \( (Δ,V,\langle \ , \ \rangle) \) be an abstract root system with Weyl group W.
     • (Hall Exer. 8.12.5) Show that the root system is simple if and only if the Weyl group acts irreducibly on it --- that is, there are no nontrivial W-invariant subspaces.
     • (Hall Exer. 8.12.6) Show that if \( \langle \cdot, \cdot \rangle' \) is another W-invariant positive definite inner product on V, then it differs from \( \langle \cdot, \cdot \rangle \) by a positive scalar.
  4. Given a root system Δ, with α ,β, α + β ∈ Δ, let β - q α, . . . , β + p α be the α-string of β. Recall the equation $$ p \cdot \frac{\langle \alpha + \beta, \alpha + \beta \rangle}{\langle \beta, \beta \rangle} = q+1$$ that we partially established in lecture on 11/21 and 11/28. It can be rewritten $$ p \cdot \left( \frac{a_{\alpha \beta}}{a_{\beta \alpha}} + a_{\alpha \beta} + 1 \right) = q+1 $$ Now let σ ∈ Δ and apply the Weyl reflection to get α' = r_σ(α), β' = r_σ(β). Let the α'-string of β' run from β' - q' α' to β' + p'α'. Show that p'=p, q'=q, a_β'α' = a_βα, and a_α'β' = a_αβ. Conclude that the equation above remains true when transformed by r_σ.
  5. (Hall Exer. 8.12.13) Let V = Rⁿ, and Δ denote the collection of vectors of the form $$ \pm e_j \pm e_k,\ k < j \qquad \pm e_j, \ j = 1, \ldots, n \qquad \pm 2 e_j, \ j=1, \ldots, n$$ Show that it satisfies the first two axioms for a reduced root system, but not the third. It is called BC_n (Note that it is the union of B_n and C_n.)

• HW 7, due 12/14 (FINAL)
  1. For the real forms su(p,q) and sl(k,H) of sl(n,C) for n = p+q or n=2k, respectively, find
     • The involutive automorphisms of the compact real form su(n) determining them.
     • Their Cartan decompositions.
  2. Hall Exer. 8.12.11: For which rank-two root systems is -I an element of the Weyl group?
  3. Recall that o(2n+1,C) has the root system B_n, comprising, for i,j = 1, ..., n, $$ \alpha(x_1, \ldots, x_n) = \left\{ \begin{array}{c} x_i - x_j \qquad i \neq j \\ \pm(x_i + x_j) \qquad i \neq j \\ \pm x_i \end{array} \right. $$ A standard choice of positive roots is $$ \Delta^+ = \left\{ \begin{array}{c} x_i - x_j \qquad i < j \\ x_i + x_j \qquad i \neq j \\ x_i \end{array} \right. $$
     • Find the indecomposable elements of Δ⁺, and thus a basis Σ = {α₁, ..., α_n} of simple roots.
     • Use the length of the α_i-string of α_j to compute the Cartan integers a_ji and the Cartan matrix for Σ.
     • Use the values in your Cartan matrix to construct the Dynkin diagram for B_n with respect to Σ.
  4. Hall Exer. 8.12.14: Determine which of the Dynkin diagrams in Figures 8.22 and 8.23 have a nontrivial automorphism. Show that only the Dynkin diagram of D₄ has an automorphism group with more than two elements.
  5. From the Dynkin diagram for F₄, reconstruct the Cartan matrix and the root system, which comprises 48 roots, 24 positive and 24 negative. (Remark: This information can easily be found in textbooks or online, but you should work it out for yourself!)