  We show that the full matrix algebra Mat_p(C) is a U-module algebra for U =
U_q sl(2), a 2p^3-dimensional quantum sl(2) group at the 2p-th root of unity.
Mat_p(C) decomposes into a direct sum of projective U-modules P^+_n with all
odd n, 1<=n<=p. In terms of generators and relations, this U-module algebra is
described as the algebra of q-differential operators "in one variable" with the
relations D z = q - q^{-1} + q^{-2} z D and z^p = D^p = 0. These relations
define a "parafermionic" statistics that generalizes the fermionic commutation
relations. By the Kazhdan--Lusztig duality, it is to be realized in a
manifestly quantum-group-symmetric description of (p,1) logarithmic conformal
field models. We extend the Kazhdan--Lusztig duality between U and the (p,1)
logarithmic models by constructing a quantum de Rham complex of the new
U-module algebra.
