# Semidirect product of Diffeomorphism group and Weyl transformations

This is more a mathematical question but in my string theory lecture we always divide in the Polyakov path integral by $$\mathrm{Diff}\ltimes \mathrm{Weyl}$$ and I was wondering why there is the semidirect product. Is there a simple argument for the semidirect product like for the Poincare group by the composition of two transformations?

1. A group element $$(f,\Omega)~\in~\mathrm{Diff}\ltimes \mathrm{Weyl}$$ acts from the right on a metric tensor $$g$$ as $$g.(f,\Omega)~=~f^{\ast}g\cdot \Omega$$ via pullback and multiplication.
2. This leads to an outer semidirect product rule: $$(f_1,\Omega_1)\bullet (f_2,\Omega_2)~=~(f_1\circ f_2,\underbrace{\Omega_1\circ f_2}_{\equiv f_2^{\ast}\Omega_1}\cdot\Omega_2).$$
• Qmechanic, this is your turf, maybe you know a quick answer. I just had a fleeting wild thought, I can't post a Q, it's too... fuzzy. Is the OP's semidirect product isomorphic to $SU(2)\ltimes\mathbb{R}^3$? Commented Dec 16, 2023 at 20:23