Cross Product Bialgebras - Part II

This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent (co-)modular co-cyclic formulation. All known examples as for instance bi- or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory. Furthermore our construction yields various novel types of cross product bialgebras.Comment: 52 pages, LaTeX. Modified proof of the central theorem and updated references included. Accepted for publication in Journal of Algebr

Bicovariant Differential Calculi and Cross Products on Braided Hopf Algebras

We consider Hopf bimodules and crossed modules over a Hopf algebra $H$ in a braided category. They are the key-stones for braided bicovariant differential calculi and their invariant vector fields respectively, as well as for the construction of braided Hopf algebra cross products. We show that the notions of Hopf bimodules and crossed modules are equivalent. A generalization of the Radford-Majid criterion to the braided case is given and it is seen that bialgebra cross products over the Hopf algebra $H$ are precisely described by $H$-crossed module bialgebras. We study the theory of (bicovariant) differential calculi in braided abelian categories and we construct \NN_0-graded bicovariant differential calculi out of first order bicovariant differential calculi. These objects are shown to be Hopf algebra differential calculi with universal bialgebra properties in the braided \NN_0-graded category.Comment: LaTeX, 15 page

Hopf (Bi-)Modules and Crossed Modules in Braided Monoidal Categories

Hopf (bi-)modules and crossed modules over a bialgebra B in a braided monoidal category C are considered. The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). Bialgebra projections and Hopf bimodule bialgebras over a Hopf algebra in C are found to be isomorphic categories. As a consequence a generalization of the Radford-Majid criterion for a braided Hopf algebra to be a cross product is obtained. The results of this paper turn out to be fundamental for the construction of (bicovariant) differential calculi on braided Hopf algebras.Comment: uuencoded compressed postscript file, 20 page

Cross Product Bialgebras - Part I

The subject of this article are cross product bialgebras without co-cycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (co-cycle free) cross product bialgebras. Besides the universal characterization we find an equivalent (co-)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid's double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfel'd double which can be constructed as a twist of a specific cross product.Comment: 33pages, t-angles.sty file needed (in xxx.lanl). Various Examples added, to be published in Journal of Algebr

Hilbert Space Representation of an Algebra of Observables for q-Deformed Relativistic Quantum Mechanics

Using a representation of the q-deformed Lorentz algebra as differential operators on quantum Minkowski space, we define an algebra of observables for a q-deformed relativistic quantum mechanics with spin zero. We construct a Hilbert space representation of this algebra in which the square of the mass $p^2$ is diagonal.Comment: 13 pages, LMU-TPW 94-