Problem Sheet E

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Problem 1 studies the symplectic linear group (my favourite matrix Lie group).
Problem 2 gives another characterisation of one-parameter subgroups.
Problem 3 asks you identify the Lie bracket on $ \mathfrak{gl}(n)$.
Problem 4 is about the flows of $ \varphi$-related vector fields.
Problem 5 shows that the Lie bracket (or Lie derivative) measures the failure for two flows to commute.
Problem 6 asks you to show that an abelian Lie group has abelian Lie algebra (you will prove the converse on the Sheet G).
Problem 7 asks to show that an embedded Lie subgroup is automatically closed.
Problem 8 shows that the functor $G \mapsto \mathfrak{g}$ is injective on the subcategory of simply-connected Lie groups.