Cross products by braided groups and bosonization (Q1314255)

The author continues the study of certain algebraic structures living in braided monoidal (or quasitensor) categories. [See also the author's survey paper, in Advances in Hopf algebras, 55-105 (1994; Zbl 0812.18004).] Let \(H\) be a cocommutative Hopf algebra and \(B\) a Hopf algebra which is a left \(H\)-module bialgebra. For these data a cross product Hopf algebra \(B\rtimes H\) can be defined (Molnar, 1977). The author obtains two generalizations of this result: 1) a braided variant where \(H\) is a braided cocommutative Hopf algebra (braided group) and \(B\) is a (quasitriangular) Hopf algebra and both live in an arbitrary quasitensor category; 2) a bosonization construction which turns any Hopf algebra \(B\) in the braided category \(_H{\mathcal M}\) of modules over a quasitriangular Hopf algebra \(H\) into the ordinary Hopf algebra \(\text{bos}(B)=B\rtimes H\). (The origin of this term stems from physics where \(\mathbb{Z}_2\)-graded algebras etc. are called `fermionic' while ordinary ungraded ones are called `bosonic'.) Earlier the author defined in some sense an adjoint procedure of transmutation which turns an ordinary Hopf algebra with quasitriangular Hopf algebra maps into a Hopf algebra in the category \(_H{\mathcal M}\). This allows the author to obtain the bosonization theorem from the braided variant of Molnar's theorem. Using the notion of quantum braided group (or quasitriangular Hopf algebra in a braided category), introduced by the author, it is possible to consider both these theorems as special cases of a braided variant of the bosonization theorem [cf. the reviewer, ``Crossed modules and quantum groups in braided categories I, II'', preprints ITP-94-21E, ITP-94-38E, Kiev].