We know that a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps.
And I saw an example of Lie group from Wikipedia:
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The 2×2 real invertible matrices form a group under multiplication, denoted by $ \operatorname{GL}(2, \mathbf{R})$ :
$ \operatorname{GL}(2, \mathbf{R}) = \left\{A=\begin{pmatrix}a&b\\c&d\end{pmatrix}: \det A=ad-bc \ne 0\right\}.$
This is a four-dimensional noncompact real Lie group.
I want to ask how to prove the multiplication and inversion of matrices in this example is smooth?
Best Answer
You have to see that it is a submanifold of $M(2,\mathbf{R})$ (which is a vector space so we know what smooth means for this one) and that multiplication is smooth in this one; for inversion you have to see how $\det$ is smooth on $M(2,\mathbf{R})$ and similarly for $A\mapsto Com(A)^T$; then the restrictions of all of these will be smooth on $GL(2,\mathbf{R})$ and you will get what you want