Equivariant Join and Fusion of Noncommutative Algebras

We translate the concept of the join of topological spaces to the language of C 17-algebras, replace the C 17-algebra of functions on the interval [0,1] with evaluation maps at 0 and 1 by a unital C 17-algebra C with appropriate two surjections, and introduce the notion of the fusion of unital C 17-algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra P with the coacting Hopf algebra H. We prove that, if the comodule algebra P is principal, then so is the fusion comodule algebra. When C=C([0,1]) and the two surjections are evaluation maps at 0 and 1, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal G-bundle X, the diagonal action of G on the join X 17G is free

Quantum principal bundles up to homotopy equivalence

The arXiv version is a corrected version of the paper published in "The Legacy of Niels Henrik Abel", 737-748. O. A. Laudal, R. Piene (eds.), Springer-Verlag 2004International audienceWe translate some fundamental properties satisfied by topological principal bundles into the setting of Hopf-Galois extensions. The properties are: functoriality, homotopy, and triviality. The main new concept of the paper is the homotopy equivalence of Hopf-Galois extensions. We work in the particular case where the subalgebra of coinvariants is central but without any restriction on the Hopf algebras coacting on the quantum principal bundle. We examine in detail the case when the Hopf algebra is one of Sweedler or Taft's finite-dimensional Hopf algebras