nLab noncommutative vector bundle

Scope

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

• 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 schemes.

• 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.

• as an associated bundle to some sort of noncommutative principal bundles, 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.

References

• S. Majid, T. Brzeziński, Quantum group gauge theory on quantum spaces, Commun. Math. Phys. 157, 591–638 (1993); Erratum 167, 235 (1995) MR94g:58015, euclid

• Tomasz Brzeziński, On synthetic interpretation of quantum principal bundles, AJSE D - Mathematics 35(1D): 13-27, 2010 arxiv:0912.0213.

• T. Brzeziński, On modules associated to coalgebra Galois extensions, J. Algebra 215, 290–317 (1999) MR2000c:16047, doi

• Zoran Škoda, Localizations for construction of quantum coset spaces, Banach Center Publications 61, 265–298, Warszawa 2003, math.QA/0301090

• Zoran Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007, pdf, (earlier different arXiv version: math.QA/0303357)

• José Gracia-Bondia, Joseph Varilly, Héctor Figueroa, Elements of noncommutative geometry, Birkhäuser 2001. xviii+685 pp. (doi:10.1007/978-1-4612-0005-5, gBooks)

• R. Coquereaux, A. O. Garcí, R. Trinchero, Associated quantum vector bundles and symplectic structure on a quantum plane, arXiv:math-ph/9908007

• Guanglian Zhang, R.B. Zhang, Equivariant vector bundles on quantum homogeneous space, pdf; R. B. Zhang, Quantum group equivariant homogeneous vector bundles, pdf slides

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:

• Piotr M. Hajac, Rainer Matthes, Wojciech Szymanski, A locally trivial quantum Hopf fibration, Algebr. Represent. Theory 9 (2006), no. 2, 121–146.

category: noncommutative geometry