In my differential topology class we have been working with Lie Groups, and we have learned that for example:
$$\mathfrak{u}(2)=T_{\text{Id}}U(2)$$
i.e. the lie algebra of $U(2)$ is the equivalent to the tangent space at the identity of the Lie Group. This is all fine to me, but when actually calculating this I found that the $\mathfrak{U}(2)$ is all skew hermitian matrices, and I just have no idea what this means. I mean the skew hermitian matrices aren't even a subset of $U(2)$ so I don't really know how to interpret this. If anyone could just tell me what I proved and what it means that would be fantastic.
Best Answer
As a comment mentioned, tangent spaces don’t need to be able to be naturally identified with a subset of the manifold. Think of a circle and a line tangent to it. The line isn’t a subset of the circle, but it still is its (affine) tangent space.