nLab noncommutative associated bundle

Distinguished from other noncommutative bundles

Here we discuss associating a bundle with fiber $F$ to a noncommutative principal bundle with structure “group” (e.g. Hopf algebra or generalization). There are also other kinds of noncommutative vector bundles (e.g. locally free sheaves on nc spaces; duals of projective modules etc.).

Affine case

The simplest case of the noncommutative principal bundle is the case of Hopf-Galois extension.

• the total and base space are affine, i.e. represented by noncommutative algebras $P$ and $U$. There is a monomorphism of algebras $U\hookrightarrow P$ (which is viewed as a comorphism of the projection among the spectra)

• $P$ is equipped with a right coaction $P\to P\otimes H$ of a fixed Hopf algebra $H$, which is an algebra map (we say that $P$ is a right $H$-comodule algebra)

• Hopf-Galois condition is satisfied: $U\hookrightarrow P$ is a $H$-Hopf-Galois extension (typically required to be faithfully flat).

The space of global sections of the associated vector bundle with $V$ as a fiber is then simply the cotensor product with the vector space $P\Box^H V$, for which we need a left $H$-comodule structure $V\to H\otimes V$. This construction however does not give the total space of the associated bundle as a noncommutative space. In commutative algebraic case to get the total algebra we need to replace vector space by the symmetric algebra of the dual, and then do the cotensor product. This can be repeated in the noncommutative case to get some algebra, however this does not reproduce the same space of sections as the cotensor product with the underlying vector space. Instead one can take the tensor algebra of the dual of the vector space as the noncommutative analogue of the symmetric algebra. After taking the cotensor product, one gets much bigger total algebra. However, it does not reproduce the quantum examples. Sometimes one should take some sort of a q-symmetric algebra, like generalizations of quantum Manin plane, what gives a consistent result just in some special cases.

Gluing via localizations

One can glue Hopf-Galois extensions along noncommutative localizations to more global objects. For example, both the total space and the base space can be represented by noncommutative schemes. An early example with nonaffine base space is described in Zoran Škoda, “Localizations for construction of quantum coset spaces” (most proofs given elsewhere). This is an application of using descent in noncommutative algebraic geometry. To get to the sheaves of sections of associated bundles, one defines the sections as a cotensor product locally on affine pieces and then glues. A simple case is treated in Zoran Škoda, “Coherent states for Hopf algebras”.

Literature

Affine case

• 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

Nonaffine case

• 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)