# Student seminar exercises

- What are the groups ?
- Compute the dimension of .
- What are the Lie algebras of the matrix groups ?
- What is as a manifold?
- Verify that the map , indeed is rotation about the axis through the angle .
- Show that the Lie algebras of and are isomorphic (as Lie algebras).
- Construct the double coverings , and .
- Consider the action of on . Find a decomposition of into the direct sum of two stable subspaces.
- Explain the picture of . What is the simply connected covering group of ?
- Show that the space of complex structures on compatible with the inner product is isomorphic to the isotropic Grassmannian of .
- For the sphere , find an isometry which reverses geodesics through . Let be the isometry group of . Consider the automorphism , where is a fixed point on the sphere. What is the fixed point subgroup ? What is the stabiliser group of in ? What is the relation between and ?
- Show that the positive-definite symmetric matrices form an open subset of the vector space of symmetric matrices.
- Prove that an open convex subset in is homeomorphic to .
- Describe the polar decomposition of . That is, every invertible complex matrix has a unique factorisation into a positive-definite Hermitian matrix and a unitary matrix (or the other way around).
- Describe the analogous decomposition in Theorem 4.2 for .
- Describe the cell decomposition of explicitly for small .
- Find a maximal torus for the Lie groups .
- Define a smooth structure on , the projective space , the Grassmannian .
- Find the tangent space at the identity (Lie algebras) of the Lie groups .