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\section*{noncommutative vector bundle}

\hypertarget{noncommutative_vector_bundles}{}\section*{{Noncommutative vector bundles}}\label{noncommutative_vector_bundles}

\noindent\hyperlink{scope}{Scope}\dotfill \pageref*{scope} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{scope}{}\subsection*{{Scope}}\label{scope}

There are at least 3 groups of ideas to define noncommutative vector bundles

\begin{itemize}%
\item taking Serre--Swan theorem as a definition in the noncommutative case; i.e. a finite-dimensional bundle is simply a projective module over the algebra of functions. Of course, in the algebraic case, this is satisfactory at best for the affine case, when possibly some Noetherianess assumptions are needed, even in the commutative case. Not good for e.g. for (nonaffine) [[noncommutative scheme]]s.


\item as a locally free sheaf of vector spaces; this requires some notion of topology or covers on the noncommutative base space. There are candidates of noncommutative topologies, cf. [[descent in noncommutative algebraic geometry]]. Problem: In commutative case, SGA gives a construction which as an input takes a quasicoherent module and as an output its ``underlying space''. There is no satisfactory construction of this form in noncommutative algebraic geometry, so the locally free sheaves do not have naturally defined total space in the category of noncommutative spaces.


\item as an associated bundle to some sort of [[noncommutative principal bundle]]s, typically with a Hopf algebra in the place of a structure group. Typically, the associated bundle is defined as a dual to some cotensor product locally, and in the affine case this is pretty standard. This has a problem as the locally free sheaf: if one wants to make a total space a space, one does a cotensor product with an algebra of functions on the space, not with a vector space itself, as the latter will just give the global sections or sheaf of sections at best. In fact one wants consistently both: to get the global sections locally as a cotensor product and to get the bundle of noncommutative spaces from a cotensor product at algebra level. The sections (in some category of noncommutative space) of the inclusion of algebras should reproduce the space of global sections, but good theorems of that kind are not known. See [[noncommutative associated bundle]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[S. Majid]], [[T. Brzeziński]], \emph{Quantum group gauge theory on quantum spaces}, Commun. Math. Phys. \textbf{157}, 591--638 (1993); Erratum 167, 235 (1995) \href{http://www.ams.org/mathscinet-getitem?mr=1243712}{MR94g:58015}, \href{http://projecteuclid.org/getRecord?id=euclid.cmp/1104254023}{euclid}


\item [[Tomasz Brzeziński]], \emph{On synthetic interpretation of quantum principal bundles}, AJSE D - Mathematics 35(1D): 13-27, 2010 \href{http://uk.arxiv.org/abs/0912.0213}{arxiv:0912.0213}.


\item T. Brzeziski, \emph{On modules associated to coalgebra Galois extensions}, J. Algebra \textbf{215}, 290--317 (1999) \href{http://www.ams.org/mathscinet-getitem?mr=1684158}{MR2000c:16047}, \href{http://dx.doi.org/10.1006/jabr.1998.7738}{doi}


\item Zoran \v{S}koda, \emph{Localizations for construction of quantum coset spaces}, Banach Center Publications \textbf{61}, 265--298, Warszawa 2003, \href{http://arxiv.org/abs/math.QA/0301090}{math.QA/0301090}


\item Zoran \v{S}koda, \emph{Coherent states for Hopf algebras}, Letters in Mathematical Physics \textbf{81}, N.1, pp. 1-17, July 2007, \href{http://www.irb.hr/korisnici/zskoda/skodacohstates.pdf}{pdf}, (earlier different arXiv version: \href{http://arxiv.org/abs/math.QA/0303357}{math.QA/0303357})


\item Jos\'e{} M. Gracia-Bond\'i{}a, Joseph C. V\'a{}rilly, H\'e{}ctor Figueroa, \emph{Elements of noncommutative geometry}, Birkh\"a{}user 2001. xviii+685 pp. \href{http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false}{gBooks}


\item R. Coquereaux, A. O. Garc\'i{}, R. Trinchero, \emph{Associated quantum vector bundles and symplectic structure on a quantum plane}, \href{http://arxiv.org/pdf/math-ph/9908007}{arXiv:math-ph/9908007}


\item Guanglian Zhang, R.B. Zhang, \emph{Equivariant vector bundles on quantum homogeneous space}, \href{http://www.maths.usyd.edu.au/u/rzhang/MathResLett.pdf}{pdf}; R. B. Zhang, \emph{Quantum group equivariant homogeneous vector bundles}, \href{http://www.newton.ac.uk/programmes/NCG/seminars/122114001.pdf}{pdf slides}



\end{itemize}
The following article is working with local triviality in the sense of ideals instead of localizations, what unfortunately corresponds to the covers by closed sets, not open:

\begin{itemize}%
\item Piotr M. Hajac, Rainer Matthes, Wojciech Szymanski, \emph{A locally trivial quantum Hopf fibration}, Algebr. Represent. Theory \textbf{9} (2006), no. 2, 121--146.

\end{itemize}
category: noncommutative geometry [[!redirects noncommutative vector bundle]] [[!redirects noncommutative vector bundles]] [[!redirects non-commutative vector bundle]] [[!redirects non-commutative vector bundles]] [[!redirects quantum vector bundle]] [[!redirects quantum vector bundles]]



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