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

TL;DR

This paper constructs a U-module algebra for a quantum group at an even root of unity, revealing a new algebraic structure with parafermionic relations and extending duality to logarithmic conformal field models.

Contribution

It introduces a new U-module algebra with parafermionic relations and extends Kazhdan--Lusztig duality to (p,1) logarithmic models.

Findings

Mat_p(C) decomposes into projective U-modules with all odd n

The algebra of q-differential operators exhibits parafermionic relations

A quantum de Rham complex is constructed for the new U-module algebra

Abstract

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.