A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

Abstract

For a Hopf algebra B, we endow the Heisenberg double ${\mathcal{H}(B^*)}$ with the structure of a module algebra over the Drinfeld double ${\mathcal{D}(B)}$. Based on this property, we propose that ${\mathcal{H}(B^*)}$ 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 ${\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}$ quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair ${(\mathcal{D}(B),\mathcal{H}(B^*))}$ is “truncated” to ${(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}$, where ${\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}$ is a ${\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}$ module algebra that turns out to have the form ${\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}$, where ${\mathbb{C}_{\mathfrak{q}}[z,\partial]}$ is the ${\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}$-module algebra with the relations z ^p = 0, ∂^p = 0, and ${\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}$.