Abstract. A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H -equivariantly projective (faithfully flat) over the coaction-invariant subalgebra P coH . We prove that principality is a piecewise property: given N comodule-algebra surjections P ! P i whose kernels intersect to zero, P is principal if and only if all P i 's are principal. Furthermore, assuming the principality of P , we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with P coH . Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N -families of surjections P ! P i and such that the comodule algebra of global sections is P . (2010). 58B32.
Mathematics Subject Classification
The Margolus-Levitin lower bound on the minimal time required for a state to be transformed into an orthogonal state is generalized. It is shown that for some initial states the new bound is stronger than the Margolus-Levitin one.
Abstract. Two hierarchies of quantum principal bundles over quantum real projective spaces are constructed. One hierarchy contains bundles with U (1) as a structure group, the other has the quantum group SU q (2) as a fibre. Both hierarchies are obtained by the process of prolongation from bundles with the cyclic group of order 2 as a fibre. The triviality or otherwise of these bundles is determined by using a general criterion for a prolongation of a comodule algebra to be a cleft Hopf-Galois extension.
We find multipullback quantum odd-dimensional spheres equipped with natural U (1)-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the K-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these K-groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces. H
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