nLab noncommutative principal bundle

Redirected from "noncommutative torsor".

There are many approaches to the generalizations of principal bundles to various flavours of noncommutative geometry. Not only that the base and total space of a principal bundle are replaced by noncommutative spaces, but various frameworks of noncommutative geometry also allow that the structure group be replaced by some analogue or generalization, something like quantum group. Some people hence talk about “quantum principal bundles”.

In noncommutative algebraic geometry, the most studied is the case in which the base and total space are affine, i.e. each represented by a single algebra, say base by $U$ and the total space by $E$ . If the structure group is a Hopf algebra, then the standard requirement is that $E$ is a right $H$ -comodule algebra which is a Hopf-Galois extension of $U$ (for algebraic story, see the references there). An affine case of connections on principal bundles is considered in

Tomasz Brzeziński, Shahn Majid, Quantum group gauge theory on quantum spaces, Commun.Math. Phys. 157, 591–638 (1993) doi

One generalization of this picture are the “coalgebra bundles”

T. Brzeziński, S. Majid, Coalgebra bundles, Comm. Math. Phys. 191 (1998), no. 2, 467–492 (arXiv version).

In that case, the Hopf algebra is replaced by a coalgebra $C$ , the total space by an algebra which is a $C$ -comodule, bu then an entwining structure is needed as an additional structure, and again a version of a Galois condition is required. The entwining which is a mixed distributive law is in fact in a role of lifting certain induced action of a monoidal category associated to $C$ from the base ground scheme cf.

Z. Škoda, Distributive laws for actions of monoidal categories, math.CT/0406310.

The liftings can be defined more generally in nonaffine situations leading to the concept of geometrically admissible actions as in

Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770.

The Galois condition can then be defined locally on some compatible cover. A generalized version of the Schneider’s descent theorem hold in this generality.

As a particular case, this allows the $H$ -comodule algebras which are not Hopf-Galois extension of their coinvariants, but became so when localizing on some affine coaction compatible cover by affine “biflat” localizations (see here). In other words the localized algebras are Hopf-Galois extensions (for example crossed product algebras) of the localized coinvariants as in

Z. Škoda, Localizations for construction of quantum coset spaces, math.QA/0301090, Banach Center Publ. 61, pp. 265–298, Warszawa 2003;
Z. Škoda, Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357).

A principal bundle is called locally trivial if there is a cover on which the Hopf-Galois extensions are in fact Hopf smash products. In the commutative case, and with affine algebraic group as a structure group, this is the same as the local triviality in fpqc topology.

Another point of view to generalized Galois conditions in noncommutative algebraic geometry based on spaces represented by monoidal categories can be found in

Tomasz Maszczyk, Noncommutative geometry through monoidal categories, math.QA/0611806

As Hopf algebroids generalize (function algebras on) groupoids, there is a well motivated study of Galois conditions (hence torsors) in the world of Hopf algebroids:

Gabriella Böhm, Galois theory for Hopf algebroids, Annali dell’Universita di Ferrara, Sez VII, Sci. Mat., Vol. LI (2005) 233–262 math.QA/0409513