Problem Sheet E

This Problem Sheet is based on Lectures 8 and 9.

• Problem 1 asks you to prove that if $X$ is $ \varphi$-related to $Y$ then $ \theta_t^X \circ \varphi  = \varphi \circ \theta_t^Y$.
• Problem 2 shows that the Lie bracket (or Lie derivative) measures the failure for two flows to commute.
• Problem 3 studies the symplectic linear group (my favourite matrix Lie group).
• Problem 4 asks to show that an embedded Lie subgroup is automatically closed.
• Problem 5 asks you to show that an abelian Lie group has abelian Lie algebra (you will prove the converse on the next sheet).
• Problem 6 asks you identify the Lie bracket on $ \mathfrak{gl}(n)$.
• Problem 7 is a bit out of place… It will be useful in lectures to come.