Quantum $s\ell(2)$ action on a divided-power quantum plane at even roots of unity

We describe a nonstandard version of the quantum plane in which the basis is given by divided powers at an even root of unity $\mathfrak q=e^{i\pi/p}$. It can be regarded as an extension of the "nearly commutative" algebra $\mathbb C[X,Y]$ with $XY=(-1)^pYX$ by nilpotents. For this quantum plane, we construct a Wess–Zumino-type de Rham complex and find its decomposition into representations of the $2p^3$-dimensional quantum group $\overline{\mathcal U}_{\mathfrak q}s\ell(2)$ and its Lusztig extension $\boldsymbol{\mathcal U}_{\mathfrak q}s\ell(2)$; we also define the quantum group action on the algebra of quantum differential operators on the quantum plane.