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Given space $\mathbb{R}^{1,d-1}$($d\ge3$), the total conformal group is $SO(d,2)$ generated by $1$-dilation, $d$-translation, $d$-special conformal, $d(d-1)/2$-Lorentz transformation.

But we know generators of Poincare symmetry and dilation are closed, so form a Lie subalgebra of Lie algebra $so(d,2)$. So what's the Lie group corresponding to Lie algebra generated only by Poincare symmetry and dilation? Is it a classical Lie group or (semi-)direct product of classical Lie group or others?

The only nontrivial commutator between dilation and Poincare symmetry is $$[D,P_\mu]=iP_\mu$$

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This is sometimes called the Weyl Group $$W(1,d-1)\sim\mathbb R \ltimes\mathrm{ISO}(1,d-1)$$

The only references I've been able to find in a quick google search are At the Frontier of Spacetime: Scalar-Tensor Theory, Bells Inequality, Machs Principle..., section 4.1.4, and Theoretical and Observational Cosmology, section 5.2. They are both on Google Books.

Indeed, dilatations are also called Weyl transformations. Unfortunately, this terminology is not universally accepted. I don't know what is the name of this group in the mathematical literature, nor whether it has been studied in detail. It doesn't seem that interesting on its own, but who am I to say.

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  • $\begingroup$ I think by Weyl transformations one usually means the more general transformations, when one rescales the metric by a scalar function. (In this case these are not coordinate transformations, but just the "manual" transformations of the metric.) $\endgroup$ – Peter Kravchuk Nov 11 '17 at 4:30

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