Wikipedia:Reference desk/Archives/Mathematics/2015 June 4

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June 4

Inverse map in lie groups

Hello,

I was reading about Lie groups, and came across the claim that the smoothness of the inverse map can be derived from the smoothness of the multiplication map. I've been thinking about this for a while but haven't gotten anywhere. Does a quick proof of this exist, or does it take a lot of machinery?

Neuroxic (talk) 11:31, 4 June 2015 (UTC)

It comes from the implicit function theorem. Let $f(x,y)=xy$. Then $x^{-1}$ is the solution of $f(x,y)=e$. By Sard's theorem, the rank of $f:G\times G\to G$ is equal to dim G in a fiber over a typical point in G. By translating by an element of G, the rank of f is constant. In particular, it is equal to dim G in the fiber over the identity. So, by the implicit function theorem, the solution y of the equation $f(x,y)=e$ depends smoothly on x. (The use of Sard's theorem is optional, but simplifies the argument somewhat.) Sławomir Biały (talk) 11:50, 4 June 2015 (UTC)

Nice! Although I feel Sard's Theorem is heavy going. What do you mean by saying its use is optional? Neuroxic (talk) 15:49, 4 June 2015 (UTC)

You can argue directly that the rank of f is constant over all of $G\times G$. Sławomir Biały (talk) 15:53, 4 June 2015 (UTC)