scholarly journals Deformation quantization of principal bundles

2016 ◽  
Vol 13 (08) ◽  
pp. 1630010
Author(s):  
Paolo Aschieri
Keyword(s):  
Quantum Group ◽  
Deformation Quantization ◽  
Principal Bundle ◽  
Base Space ◽  
Structure Group ◽  
Galois Extensions ◽  
Principal Bundles ◽  
Group Of Automorphisms ◽  
Drinfeld Twist

We outline how Drinfeld twist deformation techniques can be applied to the deformation quantization of principal bundles into noncommutative principal bundles and, more in general, to the deformation of Hopf–Galois extensions. First, we twist deform the structure group in a quantum group, and this leads to a deformation of the fibers of the principal bundle. Next, we twist deform a subgroup of the group of automorphisms of the principal bundle, and this leads to a noncommutative base space. Considering both deformations, we obtain noncommutative principal bundles with noncommutative fiber and base space as well.

scholarly journals Poisson Principal Bundles

2021 ◽  
Author(s):  
Shahn Majid ◽  
◽  
Liam Williams ◽  
Keyword(s):  
Quantum Group ◽  
Lie Group ◽  
Poisson Structure ◽  
Differential Calculus ◽  
Principal Bundle ◽  
Poisson Manifold ◽  
Spin Connection ◽  
Poisson Geometry ◽  
Hopf Fibration ◽  
Principal Bundles

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space X is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the base has an inherited Poisson structure and Poisson-compatible contravariant connection. The latter are known to be the semiclassical data for a quantum differential calculus. The theory is illustrated by the Poisson level of the q-Hopf fibration on the standard q-sphere. We also construct the Poisson level of the spin connection on a principal bundle.

scholarly journals Connections on a Parabolic Principal Bundle, II

2009 ◽  
Vol 52 (2) ◽  
pp. 175-185 ◽  
Author(s):  
Indranil Biswas
Keyword(s):  
Exact Sequence ◽  
Principal Bundle ◽  
Principal Bundles

AbstractIn Connections on a parabolic principal bundle over a curve, I we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

scholarly journals n-Tuple principal bundles

2018 ◽  
Vol 15 (12) ◽  
pp. 1850211
Author(s):  
Katarzyna Grabowska ◽  
Janusz Grabowski
Keyword(s):  
Lie Group ◽  
Compatibility Condition ◽  
Principal Bundle ◽  
Principal Bundles

We develop the concept of a double (more generally [Formula: see text]-tuple) principal bundle departing from a compatibility condition for a principal action of a Lie group on a groupoid.

HOLOMORPHIC PRINCIPAL BUNDLES WITH AN ELLIPTIC CURVE AS THE STRUCTURE GROUP

2008 ◽  
Vol 05 (06) ◽  
pp. 851-862 ◽  
Author(s):  
INDRANIL BISWAS
Keyword(s):  
Elliptic Curve ◽  
Real Number ◽  
Complex Manifold ◽  
Positive Real Number ◽  
Differential Forms ◽  
Structure Group ◽  
Positive Real ◽  
Principal Bundles ◽  
Complex Torus ◽  
Dimension One

Let Λ ⊂ ℂ be the ℤ-module generated by 1 and [Formula: see text], where τ is a positive real number. Let Z := ℂ/Λ be the corresponding complex torus of dimension one. Our aim here is to give a general construction of holomorphic principal Z-bundles over a complex manifold X. Let θ1 and θ2 be two C∞ real closed two-forms on X such that the Hodge type (0, 2) component of the form [Formula: see text] vanishes, and the elements in H2(X, ℂ) represented by θ1 and θ2 are contained in the image of H2(X, ℤ). For such a pair we construct a holomorphic principal Z-bundle over X. Conversely, given any holomorphic principal Z-bundle EZ over X, we construct a pair of closed differential forms on X of the above type.

scholarly journals Bundles over Quantum RealWeighted Projective Spaces

Axioms ◽  
2012 ◽  
Vol 1 (2) ◽  
pp. 201-225 ◽  
Author(s):  
Tomasz Brzeziński ◽  
Simon A. Fairfax
Keyword(s):  
Principal Bundle ◽  
Algebraic Approach ◽  
Projective Spaces ◽  
Projective Planes ◽  
Positive Case ◽  
Projective Modules ◽  
Principal Bundles ◽  
Seifert Manifold ◽  
Negative Case ◽  
Definition Of

The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

scholarly journals A LOOP SPACE FORMULATION FOR GEOMETRIC LIFTING PROBLEMS

2011 ◽  
Vol 90 (1) ◽  
pp. 129-144 ◽  
Author(s):  
KONRAD WALDORF
Keyword(s):  
Riemannian Manifold ◽  
Loop Space ◽  
Principal Bundle ◽  
Structure Group ◽  
Loop Spaces ◽  
Spin Structures ◽  
Space Formulation ◽  
Complex Spin ◽  
Bundle Gerbes ◽  
New Formulation

AbstractWe review and then combine two aspects of the theory of bundle gerbes. The first concerns lifting bundle gerbes and connections on those, developed by Murray and by Gomi. Lifting gerbes represent obstructions against extending the structure group of a principal bundle. The second is the transgression of gerbes to loop spaces, initiated by Brylinski and McLaughlin and with recent contributions of the author. Combining these two aspects, we obtain a new formulation of lifting problems in terms of geometry on the loop space. Most prominently, our formulation explains the relation between (complex) spin structures on a Riemannian manifold and orientations of its loop space.

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