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Is there a group $G$ with the property that $G$ is a smooth manifold, the multiplication map of $G$ is smooth, but the inversion map of $G$ is not smooth?

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I believe I've heard there are infinite-dimensional groups for which each left multiplication $g\cdot$ is smooth, but the right multiplications $\cdot g$ aren't all smooth. –  Allen Knutson Mar 16 '13 at 17:40
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For a compact manifold $M$, the group $Diff_{C^k}(M)$ (for $k\ge1$) of all $C^k$-diffeomorphisms of $M$ is a smooth manifold. Right translations are smooth, but left translations are only continuous. Inversion is only continuous. The same holds for the group $Diff_{H^s}(M)$ (for $s\ge \frac{\dim(M)}2 +1$), the Sobolev completion of order $s$ of the group of all diffeomorphisms. –  Peter Michor Mar 17 '13 at 7:05

2 Answers 2

up vote 16 down vote accepted

Bryant requires in the definition of a Lie group only that the multiplication map be smooth, and then proves that the inversion map must be smooth also. (Proposition 1, page 14.)

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Thanks Steve,most of books are gabby,it's sufficient to show that the map G to G*G s,t a to (a,-2a) is smooth. –  R Salimi Mar 16 '13 at 18:38
    
But surely (using your additive notation) proving $a\mapsto -2a$ smooth is just as hard as proving $a\mapsto -a$ smooth. –  Steven Landsburg Mar 16 '13 at 19:52
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One could add that the main ingredient in the proof is the Implicit Function Theorem. It's kind of frustrating that this property of Lie groups motivates some people to give the wrong definition of a Lie group. Of course in the end it's all the same, but the concept is wrong (and breaks down for group objects in other categories, for example $\mathsf{Top}$). –  Martin Brandenburg Mar 16 '13 at 20:27

Every Čech-complete paratopological group is a topological group. That means that for Čech-complete groups you do not have to require the continuity of the inverse, continuity of multiplication suffices. Every manifold is Čech-complete. Using the affirmative answer to Hilbert’s fifth problem we get that every paratopological group on a manifold is actually a Lie group uniquely determined by the topological group structure.

In the spirit of Martin let me give a correct (I hope I have not forgotten anything) definition which is even wronger than the definition without the inverse:

A topological space $G$ with a function $\cdot\colon G^2\to G$ is called an $n$-dimensional Lie group if and only if

  • $G$ is second-countable

  • There exists an injective, open continuous map $\iota\colon \mathbb{R}^n\to G$

  • For every $g\in G$ the map $x\mapsto g\cdot x$ is continuous and surjective

  • There exists $e\in G$ such that $x\mapsto x\cdot e$ is the identity

  • For every $g\in G$ the map $x\mapsto x\cdot g$ is continuous

  • $\cdot$ is associative

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