Student seminar exercises

  1. Some exercises from Yuhan
  2. Determine the dimension of H_k, the space of spherical harmonics of degree k on S^{n-1}.
  3. Write down some examples of spherical harmonics.
  4. Explain the decomposition P_k=\hat{H}_k\oplusr^2\hat{H}_{k-2} \oplusr^4\hat{H}_{k-4}\oplus\cdots, or equivalently, \mathbb{C}[x_1,\ldots,x_n]=\oplus_{k\geq 0}E_k\otimes H_k.
  5. Explain why E_k is an irreducible representation of \mathfrak{sl}_2(\mathbb{R}).

2022 S2

  1. Describe the Bruhat decomposition for GL_n\mathbb{C} explicitly for small n.
  2. Explain why almost all g\in GL_n\mathbb{C} have a unique factorisation g=n^-b, where n^-\in N^- and b\in B. Here n^-\in N^- is the subgroup of lower-triangular matrices with 1’s on the diagonal, and B is the subgroup of upper-triangular matrices.
  3. List a few examples of locally convex, complete, topological vector spaces.
  4. Explain the completed direct sum in Theorem 6.4.
  5. Explain why the Heisenberg group N/Z is not a matrix group.
  6. Explain why the generic fiber of the map T\times G/T\to G consists of |W| many points in the case of G=U(n) and in general.
  7. Compute the Jacobian of the map T\times G/T\to G in the case of G=U(n) and in general.

2022 S1

  1. What are the groups E_2,E_3?
  2. Compute the dimension of SO(n,\mathbb{R}).
  3. What are the Lie algebras of the matrix groups GL(n,\mathbb{R}),SL(n,\mathbb{R}), SO(n,\mathbb{R})?
  4. What is SO(3,\mathbb{R}) as a manifold?
  5. Verify that the map T_g, g=\cos\theta+u\sin\theta indeed is rotation about the axis u through the angle 2\theta.
  6. Show that the Lie algebras of SU_2 and SO_3 are isomorphic (as Lie algebras).
  7. Construct the double coverings SU_2\times SU_2\to SO_4, SL_2\mathbb{C}\to SO_{1,3}^+,\ SL_2\mathbb{R}\to SO_{1,2}^+ and SO_4\to SO_3\times SO_3.
  8. Consider the action of SO(4) on \wedge^2\mathbb{R}^4. Find a decomposition of \wedge^2\mathbb{R}^4 into the direct sum of two stable subspaces.
  9. Explain the picture of SL_2\mathbb{R}. What is the simply connected covering group of SL_2\mathbb{R}?
  10. Show that the space of complex structures on \mathbb{R}^{2n} compatible with the inner product is isomorphic to the isotropic Grassmannian of \mathbb{C}^{2n}.
  11. For the sphere S^{n-1}, find an isometry f_x which reverses geodesics through x. Let G=O(n) be the isometry group of S^{n-1}. Consider the automorphism \alpha:G\to G,\,g\mapsto f_{x_0}\circ g\circ f_{x_0}, where x_0 is a fixed point on the sphere. What is the fixed point subgroup G^\alpha? What is the stabiliser group H of x_0 in G? What is the relation between G^\alpha and H?
  12. Show that the positive-definite symmetric matrices form an open subset of the vector space of symmetric matrices.
  13. Prove that an open convex subset in \mathbb{R}^n is homeomorphic to \mathbb{R}^n.
  14. Describe the polar decomposition of GL(n,\mathbb{C}). That is, every invertible complex n\times n matrix has a unique factorisation into a positive-definite Hermitian matrix and a unitary matrix (or the other way around).
  15. Describe the analogous decomposition in Theorem 4.2 for GL(n,\mathbb{R}).
  16. Describe the cell decomposition of GL_n\mathbb{C}/B explicitly for small n.
  17. Find a maximal torus for the Lie groups SO_{n}\mathbb{R}, U_n,SU_n,Sp_{2n}\mathbb{R}.
  18. Define a smooth structure on O_3, the projective space \mathbb{P}_{\mathbb{R}}^n, the Grassmannian Gr_k(\mathbb{R}^n).
  19. Find the tangent space at the identity (Lie algebras) of the Lie groups SO_{n}\mathbb{R}, U_n,SU_n,Sp_{2n}\mathbb{R}.