A note on the differential smoothness of skew PBW extensions
Andrés Rubiano, Armando Reyes
TL;DR
This work addresses when differential smoothness transfers from a coefficient ring $R$ to a skew PBW extension $A=\sigma(R)\langle x_1,\dots,x_n\rangle$. It develops a framework of extended automorphisms and derivations $\widetilde{\sigma}_i,\widetilde{\delta}_i$ and proves a main theorem: if the system $\{\sigma_i\}$ is commutative, the cross-relations satisfy $d_{i,j}=1$ and $r_k^{(i,j)}=0$, and the extensions respect compatibility conditions, then $A$ is differentially smooth; its GK-dimension equals $n$ and an $n$-dimensional connected integrable calculus with a volume form $\omega$ exists. The construction leverages the Brzeziński-Sitarz calculus, defining $\Omega^1(A)$ with generators $dx_i$ and actions twisted by the extended automorphisms, and verifying integrability via explicit duality isomorphisms. This provides a concrete pathway to extend differential smoothness from $R$ to SPBW extensions, generalizing earlier results for Ore extensions to the SPBW setting.
Abstract
We investigate the differential smoothness of a certain family of skew Poincaré-Birkhoff-Witt extensions.