Bosonisations and Di erentials on Inhomogeneous Quantum Groups
Abstract
We dualise Majid's double bosonisation to nd a construction of coquasitriangular Hopf Bop>/ A .<B which we call codouble bosonisation, where B is a nite-dimensional braided Hopf algebra living in the category of comodules over coquasitriangular Hopf algebra A. We then construct a reduced quantum coordinate algebra cq[SL2] at q primitive n-th of unity by codouble bosonisation and nd new generators for cq[SL2] such that their monomials are essentially a dual basis to the standard PBW basis of the reduced Drinfeld-Jimbo quantum enveloping algebra uq(sl2). Our methods apply in principle for general cq[G] as we illustrate for the case of cq[SL3] at certain odd roots of unity. We also introduce a method of nding di erential calculi on double cross product A./H, biproduct A .<B, and bicrossproduct AI/H Hopf algebras by constructing their super version. We apply our method to construct the natural di erential calculus on the generalised quantum double D(A;H) = Aop./H such that the resulting exterior algebra acts di erentiably on H, and on the double coquasitriangular Hopf algebras A./RA such that the resulting exterior algebra acts and coacts di erentiably on A. We also construct (Cq[GL2.<C2]) for the quantum group of a ne transformation of the plane and (Poinc1;1) for the bicrossproduct Poincar e group in 2 dimensions such that the resulting exterior algebras are strongly bicovariant and coact di erentiably on the canonical comodule algebras associated to these inhomogeneous quantum groups.
Authors
Aziz, Ryan KasyfilCollections
- Theses [4144]