Ch 7: 13, Ch 8: 19,22,28,31

For each $n \geq 􏰆 1$, prove that $U(n)$ is a properly embedded $n^2$-dimensional Lie subgroup of $GL(n, \C)$.

We follow the route of proof in Example 7.27 (page 166-167 of Lee). First, we construct a map from $U(n)$ to $M(n, \C)$, that maps $$\Phi: GL(n, \C) \to M(n, \C), \quad A \mapsto A^*A. $$ Then, $GL(n,\C)$ acts on the right of $M(n, \C)$ by conjugation, $(g, M) \mapsto g^{*} M g$. This is a right action of $U(n)$ on $M$. Then, by equivariant constant rank theorem, we conclude that $\Phi^{-1}(I)$ is properly embedded submanifold.

Show that $\R^3$ with the cross product is a Lie algebra.

Just need to check the Jacobi identity on the generators $i,j,k$.

Derivation of an associated algebra forms a Lie algebra.

Just check

Show that determinant map's derivative is the trace maps. Hint: 7-4(a)

Ideal of a Lie algebra.