2001/01/05 by Florin Panaite, Panaite, Florin
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.math/0101039
30 pages, Latex, no figures
arxiv created 2001/01/05 · arxiv updated 2009/11/30
Let A be a finite dimensional Hopf algebra and (H, R) a quasitriangular bialgebra. Denote by H^*R a certain deformation of the multiplication of H^* via R. We prove that H^*R is a quantum commutative left H⊗ Hop cop-module algebra. If H is the Drinfel'd double of A then H^*R is the Heisenberg double of A. We study the relation between H^*R and Majid's "covariantised product". We give a formula for the canonical element of the Heisenberg double of A, solution to the pentagon equation, in terms of the R-matrix of the Drinfel'd double of A. We generalize a theorem of Jiang-Hua Lu on quantum groupoids and using this and the above we obtain an example of a quantum groupoid having the Heisenberg double of A as base. If, in Richard Borcherds' concept of a "vertex group" we allow the "ring of singular functions" to be noncommutative, we prove that if A is a finite dimensional cocommutative Hopf algebra then the Heisenberg double of A is a vertex group over A. The construction and properties of H^*R are given also for quasi-bialgebras and a definition for the Heisenberg double of a finite dimensional quasi-Hopf algebra is proposed.