A Heisenberg Double Addition to the Logarithmic Kazhdan-Lusztig Duality

作者：Semikhatov Alexei M^*

来源：Letters in Mathematical Physics, 2010, 92(1): 81-98.

DOI：10.1007/s11005-010-0373-9

• Drinfeld double
• Heisenberg double
• module algebras
• quantum sl(2)
• roots of unity

For a Hopf algebra B, we endow the Heisenberg double H(B*) with the structure of a module algebra over the Drinfeld double D(B). Based on this property, we propose that H(B*) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan-Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the (U) over bar qsl(2) quantum group that is Kazhdan-Lusztig-dual to (p, 1) logarithmic conformal models. The corresponding pair (D(B), H(B*)) is "truncated" to ((U) over bar qsl (2), (H) over bar qsl (2)), where (H) over bar qsl(2) is a (U) over bar qsl(2) module algebra that turns out to have the form (H)over bar (q)sl(2)=C(q)[z,partial derivative]circle times C[lambda]/(lambda(2)p-1), where C(q)[z,partial derivative] is the (U) over bar (q)sl(2)-module algebra with the relations z(p) = 0, partial derivative(p) = 0, and partial derivative(z)= q-q(-1) + q(-2)z partial derivative.

• 出版日期2010-4