First, I want to say that I posted this in the physics forum but no one seems to be interested to respond so because I really believe that my question can be figured out by mathematicians Am here.
When studying Poincaré gauge theory using Milutin Blagojevich's book on "Gravitation and gauge symmetries" we find an interesting line of thought. But to get that I need to set some equations first.
When doing the Poincaré gauge theory, i.e. taking global transformations making them local and doing the Gauge procedure by adding new fields we find The rotational gauge field $A^{ij}_{\ \ \ \ \mu}$ and the translational gauge field $h_k^{\ \ \mu}$.
The consistency of the theory needs to be maintained by imposing a certain variation on the gauge fields $$\delta h_{k}{}^{\mu}=w_{k}{}^{i} h_{i}{ }^{\mu}-\xi^{\nu} \partial_{v} h_{k}{}^{\mu}+h_{k}{ }^{v} \partial_{\nu} \xi^{ \mu}$$ and $$\delta A^{i j}{}_{\mu}=-\partial_{\mu} w^{i j}+A_{k}{}^{j}{}_{\mu} w^{i k}+A^{i}{}_{k\mu} w^{jk}-\left(\partial_{\mu} \xi^{\nu}\right) A^{i j}{}_{\nu}-\xi^{\nu} \partial_{\nu} A^{i j}{}_{\mu}$$
These relations will maintain the Lagrangian invariant even when passing to local transformations.
On the other hand, in a more mathematical way in the book he says
It is an intriguing fact that PGT does not have the structure of an ‘ordinary’ gauge theory (McDowell and Mansouri 1977, Regge 1986, Banados et al 1996). To clarify this point, we start from the Poincaré generators $P_a$, $M_{ab}$ satisfying the Lie algebra (2.6), and define the gauge potential as $A_\mu=e^a{}_\mu P_a + \frac{1}{2} \omega^{ab}{}_\mu M_{ab}$. The infinitesimal gauge transformation $$\delta_{0} A_{\mu}=-\nabla_{\mu} \lambda=-\partial_{\mu} \lambda-\left[A_{\mu}, \lambda\right]$$ where $\lambda=\lambda^{a} P_{a}+\frac{1}{2} \lambda^{a b} M_{a b}$, has the following component content:\begin{aligned} \text { Translations: } & \delta_{0} e^{a}{ }_{\mu}=-\nabla_{\mu}^{\prime} \lambda^{a} & & \delta_{0} \omega^{a b}{}_{\mu}=0 \\ \text { Rotations: } & \delta_{0} e^{a}{ }_{\mu}=\lambda^{a}{}_b e^{b}{ }_{\mu} & & \delta_{0} \omega^{a b}{ }_{\mu}=-\nabla_{\mu}^{\prime} \lambda^{a b} \end{aligned} where $\nabla' = \nabla(\omega)$ is the covariant derivative with respect to the spin connection $\omega$. The resulting gauge transformations are clearly different from those obtained in PGT.
I have questions about this result:
how did he get this infinitesimal gauge transformation $\delta_{0} A_{\mu}=-\nabla_{\mu} \lambda=-\partial_{\mu} \lambda-\left[A_{\mu}, \lambda\right]$?
maybe this one is related to the first but what is $\lambda$, and what's the point of it?
and last, how did he get these gauge field transformations and why are they separated into two parts translations and rotations?
It's difficult for me to explicit all the mathematical framework and all the gauge procedure of this subject. But if my post needs more clarity I will make changes so you can understand more.
This answer might be a bit late, but I think you can relevant details in Hamilton's Mathematical Gauge theory book and the references therein. I post the quote, which is relevant for the discussion:
References: Mark J.D. Hamilton, Mathematical Gauge Theory With Applications To The Standard Model of Particle Physics, Springer.