A differential U-module algebra for U = A(a) (q) sa""(2) at an even root of unity

作者：Semikhatov A M^*

来源：Theoretical and Mathematical Physics, 2009, 159(1): 424-447.

DOI：10.1007/s11232-009-0035-1

• quantum group
• parafermionic statistics
• U-module algebra
• Kazhdan-Lusztig duality
• logarithmic conformal field theory

We show that the full matrix algebra Mat (p)(a",) is a U-module algebra for U = A(a) (q) sa""(2), a quantum sa""(2) group at the 2pth root of unity. The algebra Mat (p)(a",) decomposes into a direct sum of projective U-modules P (n) ( ) with all odd n, 1 a parts per thousand currency sign n a parts per thousand currency sign 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 a,z = q - q (-1) q (-2) za, and z(p) = a,(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 and discussing its field theory counterpart.

• 出版日期2009-4