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A conformal linear map $A : V \to V$ is a map such that $$ \frac{\langle Av, Aw\rangle}{|\langle Av , Aw \rangle |} = \frac{\langle v, w\rangle}{|\langle v, w, \rangle |} $$ for all nonzero vectors $v,w \in V$.

Clearly the set of all conformal linear maps on $(V, \langle \cdot, \cdot \rangle)$ forms a Lie group, but I never see this group mentioned along with other Lie groups. Does this Lie group have a canonical abbreviation (like $SO(n)$ for the special orthogonal group on $n$-dimensions)?

I'm sort of surprised that this group doesn't come up often because in complex analysis for example the idea of a conformal but not necessarily orthogonal map is quite important.

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The Wikipedia article uses the abbreviation $CO(V, Q)$ for the conformal group of a vector space $V$ equipped with a quadratic form $Q$, which I've never seen before. Apparently in indefinite signature things get more interesting but if $Q$ is positive-definite it's not hard to see that a conformal $A$ is just an orthogonal $A$ up to scale, so the resulting Lie group is just an orthogonal group times $\mathbb{R}_{+}$. So its study reduces pretty much immediately to the study of the orthogonal group.

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  • $\begingroup$ So not often commented on because there isn't much to say about it that isn't already covered by the classical examples? $\endgroup$ Jun 26 at 0:01
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    $\begingroup$ @Charles: I suppose not. It's also worth knowing that there's a lot to say about conformal symmetries, especially in $2$ dimensions, with applications to physics. Here's a place to start: en.wikipedia.org/wiki/Virasoro_algebra $\endgroup$ Jun 26 at 1:05
  • $\begingroup$ I was aware of widespread applications of conformal symmetries in physics and complex analysis. That's why I was surprised that the group of conformal symmetries isn't more often commented on in introductory texts on lie groups etc. $\endgroup$ Jun 26 at 1:15

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