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

Abstract

We introduce a quasitriangular Hopf algebra or 'quantum group' U (B), the double-bosonisation, associated to every braided group B in the category of Hmodules 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 = f recovers Lusztig's construction of U q (g), where f is Lusztig's algebra associated to a Cartan datum; 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 U q (sl 3 ) from U q (sl 2 ) by this method, extending it by the quantum-braided plane A 2 q . 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.

Pointed Hopf Algebras with Triangular Decomposition

“…The definition of the asymmetric braided Drinfeld double can be given using Tannakian reconstruction theory by describing their category of modules. This is similar to the approach used for the braided Drinfeld double in [24 (2) c (1) , b (1) (2) , (28) …”

“…It was shown in [24] that U q (g) in fact is a braided Drinfeld double (which is referred to as a double bosonization there). It was proved in [14] that also two-parameter quantum groups are Drinfeld doubles.…”

“…We need two braided Hopf algebras which are only required to be dually paired considered as braided Hopf algebra in the category of modules (rather than YDmodules). That is, the requirement that is weakened compared to the definition of a braided Drinfeld double (as in [24] or [21]) is that the comodule structures do not need to be dually paired. We refer to this generalization as the asymmetric braided Drinfeld double.…”

“…This form of the Drinfeld double was introduced as the double bosonization in [23,24], see also [21, Section 3.5] for the presentation used here. We now give a more general definition of an asymmetric braided Drinfeld double which is suitable to capture the more general class of Hopf algebras that we find in Section 4, including multiparameter quantum groups, as examples.…”

“…With these structures, the relation (36) becomes Note that a partial converse statement also holds: Given an asymmetric braided Drinfeld double that is symmetric, then it can be displayed as a braided Drinfeld double in the sense of [21,24], but unless H is quasitriangular (and the coactions induced by the R-matrix), we need to view it over the base Hopf algebra Drin(H ) instead. If the positive and negative part are perfectly paired, then we can give a formal power series describing the R-matrix and an appropriate subcategory (corresponding to the Drinfeld center) is braided.…”