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Urs, David Roberts and I got into discussion of locally trivial noncommutative bundles in a discussion with a wrong title (see around here), so let us better move it here. There are still some of my latest posts there which Urs and David might have not yet seen.
I decided to update a bit noncommutative principal bundle, so I will start today a bit.
New related entries geometrically admissible action, noncommutative vector bundle, noncommutative associated bundle.
New entry noncommutative space as a cover containing a very interesting category defined by Kontsevich and Rosenberg in 1997. It would be a big thing if somebody could redefine this category as some category of “sheaves” on $NAff_k$.
I made some changes to the closely related gebra entries crossed product algebra and cleft extension (cleft comodule algebra). I should post some blog about what I am thinking behind the scene, but I am in Zeitnot so I will barely be able to do some hints. So, first of all, I am finishing some article(s) on noncommutative bundles with Hopf algebra in the role of structure group and with localizations (possibly not mutually compatible, i.e. the localization functors do not pairwise mutually “commute”) used instead of topology. This is quite well thought with a complicated but clean class of examples in hand and at least one preprint should be at arXiv in few days. On the other hand, there is something I just started thinking and is quite close to this.
Namely, unless I had some basic misconception, the Sweedler coring canonically has the additional structure map required in the formalism of Kontsevich and Rosenberg mentioned in 3. Second when we extend from the base to the total space we get sort of triviality, namely we get what is called the cleft comodule algebra. Usually we say that the trivial bundle in noncommutative geometry corresponds to the case when the cleavage is algebra map (smash product algebra) but this is a matter of taste. In my examples local triviality really gives smash product but some others have natural examples with the latter, more general case. Now the extension of scalars is controlled by Sweedler coring. I know how to write some sort of transition functions for coring with the structure map in the sense of Kontsevich-Rosenberg and this is well behaved with respect to associating bundles. Very interesting sort of Čech cohomology stems out of that and can be defined at the level of a coring. The latter also implies the gluing for the covers by closed sets from works of people working in operator algebras rather than with noncommutative localization.
Let me just motivate $\gamma:H\to E$ from the classical, commutative theory. So let $H=O(G)$ be the coordinate ring of the affine algebraic group and $E = O(P)$ the functions on the total space of a principal bundle over $U = O(X)$ which is trivial, via a section $t:X\to P$. Then any function $\phi:P\to\mathbf{R}$ can be written in terms of the horizontal and vertical coordinates of the trivialized bundle. For $e\in P$, the projection is $\pi(e)$, that is the horizontal coordinate. The vertical coordinate is relative to the section, and is obtained via the translation map $\tau:P\times_X P\to G$, $\tau:(e_1,e_2)\mapsto g$ such that $e_2 = e_1 g$. So $e = t(\pi(e))\tau(t(\pi(e)),e)$ or in relative coordinates $(\pi(e),\tau(t(\pi(e)),e)$ (as $t(x)$ corresponds to $(x,1)$ in the trivialization isomorphism).
Now functions on the product give the tensor product of functions. The recipe above tells that the splitting for the algebra of function is given by the homomorphism of right $O(G)$-comodule algebras which to a function $f:G\to\mathbf{R}$ assigns $f(\tau(t(\pi(e)),e))\in O(P) = E$.
Also let me also explain the local triviality in the sense of cleft extension of the extension of scalars to the total algebra. If we start with a Hopf-Galois extension what means that $U=E^{\mathrm{co}H}\hookrightarrow E$, and the map
$can: e\otimes e'\mapsto \sum e e'_{(0)}\otimes e'_{(1)}$is invertible where $\rho(e) = \sum e_{(0)}\otimes e_{(1)}$ is the extended Sweedler notation for coactions. After the extension of scalars along $U\to E$ we get $E\otimes_U E$ with coaction $e\otimes e'\mapsto \sum e\otimes e'_{(0)}\otimes e'_{(1)}$. Then
$\gamma: h\mapsto can^{-1}(1\otimes h)$is a comodule map. In the commutative case, this map is invertible with $\gamma^{-1} = can^{-1}(1\otimes S h)$ where $S$ is an antipode because then $can$ is a homomorphism of algebras. Unfortunately, I do not see its convolution invertibility in general. Still for locally cleft extensions on the cover one has a good theory of Čech coczcles (there are some interesting phenomena there, however, not existing in commutative case).
Maybe somebody will be interested: cleft extension of a space cover (zoranskoda).
And in this
The article is rather sketchy as it is submitted as a letter with 8 page limitation. More details will be elsewhere soon (and in greater generality).
That sounds good. The question of principal bundles in that dual Hopfy setting had occurred to me some years ago when reading up on quantum groups after a talk by Majid, but I did not know enough to try to do anything with it.
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