Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2008-08-31
Theor. Math. Phys. 159 (2009) 424--447
Physics
High Energy Physics
High Energy Physics - Theory
29 pages, amsart++, xypics. V3: The differential U-module algebra was claimed quantum commutative erroneously. This is now cor
Scientific paper
10.1007/s11232-009-0035-1
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.
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