# Name for group of conformal linear maps?

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.

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.
• @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 Jun 26 at 1:05