The Lie algebra of $U(n)$ is given by $n \times n$ skew-Hermitian matrices, with the bracket given by the usual commutator $[A,B] = AB- BA$.
I'm curious what the Lie algebra of $C_2 \ltimes U(n)$ is. In particular, I want to understand the underlying vector space of this Lie algebra, to better understand the adjoint representations of these compact Lie groups.
Does the short exact sequence $$ 0 \rightarrow U(n) \rightarrow C_2 \ltimes U(n) \rightarrow C_2 $$ induce a short exact sequence of lie algebras? And if so, does this yield that the underlying vector space for the Lie algebra of $C_2 \ltimes U(n)$ is the underlying vector space of the Lie algebra of $U(n)$, since the Lie algebra of $C_2$ is zero-dimensional?