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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 allodd n, 1<=n<=p. In terms of generators and relations, this U-module algebra isdescribed as the algebra of q-differential operators "in one variable" with therelations D z = q - q^{-1} + q^{-2} z D and z^p = D^p = 0. These relationsdefine a "parafermionic" statistics that generalizes the fermionic commutationrelations. By the Kazhdan--Lusztig duality, it is to be realized in amanifestly quantum-group-symmetric description of (p,1) logarithmic conformalfield models. We extend the Kazhdan--Lusztig duality between U and the (p,1)logarithmic models by constructing a quantum de Rham complex of the newU-module algebra.

A differential U-module algebra for U=U_q sl(2) at an even root of unity