FS-coalgebras and crossed coproducts

In this paper, we introduce FS-coalgebras, which provide solutions of FS-equations and also solution of braid equations considered by Caenepeel, Militaru and Zhu. FS-coalgebras are constructed by using FS-equations and Harrison cocycles. As applications, we prove that every bialgebra H is an FS-bialgebra if and only if there is a two-sided integral α in {H^{\ast}} such that {\varepsilon(\alpha)=1} , and we show that the crossed coproduct {H^{R}} introduced by the Harrison cocycle R is an FS-coalgebra when {(H,R)} is a finite-dimensional quasitriangular Hopf algebra or a Long copaired bialgebra.

Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

TWISTING ALL THE WAY: FROM ALGEBRAS TO MORPHISMS AND CONNECTIONS

Given a Hopf algebra H and an algebra A that is an H-module algebra we consider the category of left H-modules and A-bimodules [Formula: see text], where morphisms are just right A-linear maps (not necessarily H-equivariant). Given a twist [Formula: see text] of H we then quantize (deform) H to [Formula: see text], A to A⋆ and correspondingly the category [Formula: see text] to [Formula: see text]. If we consider a quasitriangular Hopf algebra H, a quasi-commutative algebra A and quasi-commutative A-bimodules, we can further construct and study tensor products over A of modules and of morphisms, and their twist quantization. This study leads to the definition of arbitrary (i.e., not necessarily H-equivariant) connections on quasi-commutative A-bimodules, to extend these connections to tensor product modules and to quantize them to A⋆-bimodule connections. Their curvatures and those on tensor product modules are also determined.

QUASITRIANGULARITY OF BRZEZIŃSKI'S CROSSED COPRODUCTS

In continuation of our recent work about the quasitriangular structures for the twisted tensor biproduct, we give the necessary and sufficient conditions for Brzeziński crossed coproduct coalgebra, including the twisted tensor coproduct introduced by Caenepeel, Ion, Militaru and Zhu and crossed coproduct as constructed by Lin, equipped with the usual tensor product algebra structure to be a Hopf algebra. Furthermore, the necessary and sufficient conditions for Brzeziński crossed coproduct to be a quasitriangular Hopf algebra are obtained.

Double-bosonization of braided groups and the construction of Uq(g)

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.

On the Structure of Monodromy Algebras and Drinfeld Doubles

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double [Formula: see text]. Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov–Tian–Shansky, both algebras are isomorphic to the algebraic tensor product H ⊗ H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlachányi and the author. In the Appendix the multi-loop algebras ℒm of Alekseev and Schomerus [3] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the "bosonization formula" of [3] representing ℒm as H ⊗…⊗ H.

A METHOD FOR OBTAINING QUANTUM DOUBLES FROM THE YANG-BAXTER R-MATRICES

We develop the approaches of Faddeev, Reshetikhin and Takhtajan1 and Majid2 which enable one to associate a quasitriangular Hopf algebra to every regular invertible constant solution of the quantum Yang-Baxter equations. We show that such a Hopf algebra is actually a quantum double.