Quantum principal bundles up to homotopy equivalence

Hopf-Galois extensions are known to be the right generalizations of both Galois field extensions and principal G-bundles in the framework of non-commutative associative algebras. An abundant literature has been devoted to them by Hopf algebra specialists (see [Mg], [Sn1], [Sn2] and references therein). Recently there has been a surge of interest in Hopf-Galois extensions among mathematicians and theoretical physicists working in non-commutative geometry à la Connes and à la Woronowicz (cf. [BM1], [BM2], [Ha1], [Ha2], [HM], [Mj]). In their work HopfGalois extensions are considered in the setting of “quantum group gauge theory”. In this note we deal with Hopf-Galois extensions in the light of topology. This leads us to simple questions for which we have very few answers, but which ought to be of interest to those working on Hopf algebras and in non-commutative geometry. We derive these questions from certain fundamental properties satisfied by topological principal bundles when we translate them into the setting of HopfGalois extensions. The properties we consider are the following. (I) (Functoriality) Given a principal G-bundle X → Y and a map f : Y ′ → Y , then the pull-back fX → Y ′ is a principal G-bundle. (II) (Homotopy) If f, g : Y ′ → Y are homotopy equivalent maps, then fX and gX are homotopy equivalent bundles. (III) (Triviality) Any principal G-bundle over the point is trivial. In order to translate these properties into algebra, we introduce what we call the homotopy equivalence of Hopf-Galois extensions. This is the main new concept of this note. For our definition of homotopy equivalence we need to restrict to extensions in which the subalgebra of coinvariants is central. In other words, the bases of the quantum principal bundles we consider belong to classical (commutative) geometry. Nevertheless, we impose no restriction on the “structural groups”, that is on the Hopf algebras coacting on the quantum principal bundles: they may be non-commutative and non-cocommutative, infinite-dimensional, etc.