  We describe a nonstandard version of the quantum plane, the one in the basis
of divided powers at an even root of unity $q=e^{i\pi/p}$. It can be regarded
as an extension of the "nearly commutative" algebra $C[X,Y]$ with $X Y =(-1)^p
Y X$ 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 $U_q sl(2)$ and its Lusztig extension; the
quantum group action is also defined on the algebra of quantum differential
operators on the quantum plane.

Quantum Algebra