Noncommutative Principal Bundles Through Twist Deformation

Abstract

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

Mapping spaces and automorphism groups of toric noncommutative spaces

“…We choose a cotriangular structure on H , i.e., a linear map R: 4c) for all f, g, h ∈ H , where we have used Sweedler notation (h) = h (1) ⊗ h (2) (with summation understood) for the coproduct in H . The quasi-commutativity condition (1) ) h (2) g (2) , for all g, h ∈ H , is automatically fulfilled because H is commutative and cocommutative.…”

“…The quasi-commutativity condition (1) ) h (2) g (2) , for all g, h ∈ H , is automatically fulfilled because H is commutative and cocommutative. For example, if K = C is the field of complex numbers, we may take the usual cotriangular structure defined by 5) where is an antisymmetric real n×n-matrix, which plays the role of deformation parameters for the theory.…”

“…These results were later formalized within the powerful framework of closed braided monoidal categories and therefore generalized to certain nonassociative spaces (obtained by cochain twist deformations) in [5,6]. Examples of noncommutative principal bundles (i.e., Hopf-Galois extensions [9,10]) in this framework were studied in [20], and these constructions were subsequently abstracted and generalized in [1]. In applications to noncommutative gauge theory, moduli spaces of instantons on toric noncommutative spaces were analyzed in [7,8,11,13], while analogous moduli spaces of self-dual strings in higher noncommutative gauge theory were considered by [25].…”

Extended noncommutative Minkowski spacetimes and hybrid gauge symmetries

“…Here all possible types of (1 + 1) "hybrid gauge symmetries" and, therefore, of their associated noncommutative algebras generated by the spacetime, x, and gauge, q coordinates are obtained as the quantum homogeneous spaces arising from coisotropic quantum deformations of the centrally extended Poincaré Lie algebra. One of the cases so obtained is just the so-called canonical or θ -noncommutative spacetime (see [41][42][43][44] and references therein), that can be thus interpreted as being the quantum homogeneous space of a certain quantum Poincaré group. In Sect.…”

“…The b 1 = 0 case leads to the well-known θ -noncommutative spacetime in (1+1) dimensions (see [41][42][43][44] and references therein), which can be thus considered as an extended noncommutative Minkowski spacetime which is invariant under a certain quantum deformation of the extended (1 + 1) Poincaré group. B.2 When b 1 = 0, we obtain a 3-parameter family of deformations…”