We study infinite dimensional generalisations of the Heisenberg doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated canonical elements which satisfy pentagon equation. The former reproduces the canonical element, expressed using the Faddeev's quantum dilogarithm, which has been found by Kashaev to be realised within quantised Teichmüller theory, while for the latter we show that it corresponds to an operator from quantised super Teichmüller theory. We study infinite dimensional representations of those two Heisenberg doubles and, using an algebra homomorphism between Heisenberg doubles and Drinfeld doubles, we find associated representations of Drinfeld doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$. Moreover, we reproduce the previously obtained $R$-matrix for the former and derive a novel $R$-matrix for the latter representation.
Heisenberg and Drinfeld doubles of Uq(gl(1|1)) and Uq(osp(1|2)) super-algebras
Section of Mathematics, University of Geneva（日内瓦大学数学系）；Centre for Quantum Mathematics Department of Mathematics and Computer Science (IMADA), Southern Denmark University (SDU)（南丹麦大学数学与计算机科学系量子数学中心）；CentraleSupélec, LGPM（中央理工-高等电力学院LGPM实验室）
Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.
Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group $U_{q,x}(\widehat{sl_2})=U_{q,λ}(\widehat{sl_2})$, Drinfeld Currents and Hopf Algebroid Structures
The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has…
