Quantum group Gauge theory on quantum spaces

Brzeziński, Tomasz; Majid, Shahn

doi:10.1007/bf02099359

Cited by 165 publications

(661 citation statements)

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“…In this example A is actually a Hopf algebra andẽ(h) = q1 as here yields a bundle with is equivalent (in the coalgebra bundle version) to a Hopf algebra bundle as in [7]. On the other hand, other choices ofẽ yield algebra bundles which are not equivalent to Hopf algebra bundles, i.e.…”

Section: Galois Actions and Algebra Factorisationsmentioning

confidence: 99%

“…This is the natural analogue for coalgebra bundles of the associated bundles in the quantum group gauge theory of [7].…”

Section: (I) the Left Associated Bundle (Or Module) To A Left C-comodulementioning

confidence: 99%

“…This in turn implies that for all i, u i ∈ M. Now we can extend the notion of a strongly horizontal form from [7].…”

Section: Lemma 52 For a Copointed Coalgebra Bundle P (M C ψ E) mentioning

confidence: 99%

“…In particular, it need not be a quantum group, which would be too restrictive for many interesting examples. In [5] the theory of modules or "associated bundles" is extended to this case along the lines of the quantum group case in [7]. We apply this now to extend the recently introduced notion of a frame resolution [23], thereby bringing the coalgebra version of the gauge theory in line with the more restrictive quantum group gauge theory case.…”

Section: Introductionmentioning

confidence: 99%

“…6, we show that the coalgebra theory allows one to include the crucial example of the monopole on the full 2-parameter family of Podleś quantum spheres [28]. Recall that Podleś classified all reasonable "quantum spheres" covariant under the quantum group SU q (2), and until now the q-monopole has been constructed [7] only for a diagonal subfamily (the so-called standard quantum spheres). The general case requires the more general coalgebra bundle theory.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Brzeziński

Majid

2000

Commun. Math. Phys.

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We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M 2 (C) = CZ 2 · CZ 2 . We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S 2 q,s .

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Section: Galois Actions and Algebra Factorisationsmentioning

confidence: 99%

“…This is the natural analogue for coalgebra bundles of the associated bundles in the quantum group gauge theory of [7].…”

Section: (I) the Left Associated Bundle (Or Module) To A Left C-comodulementioning

confidence: 99%

“…This in turn implies that for all i, u i ∈ M. Now we can extend the notion of a strongly horizontal form from [7].…”

Section: Lemma 52 For a Copointed Coalgebra Bundle P (M C ψ E) mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Brzeziński

Majid

2000

Commun. Math. Phys.

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Bicrossproduct Structure of the Quantum Weyl Group

Majid¹

1994

Journal of Algebra

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We show that the affine quantum group U q (ŝl 2 ) is isomorphic to a bicrossproduct central extension CZ χ ◮⊳U q (Lsl 2 ) of the quantum loop group U q (Lsl 2 ) by a quantum cocycle χ, which we construct. We prove the same result for U q (ĝ) in R-matrix form.Keywords: affine quantum group -central extension -quantum cocycle -loop group -R-matrix.The required theory of (non-Abelian) Hopf algebra extensions was already introduced (by the author) in [4], where it is shown that extensions generally have the form of a cocycle bicrossproduct. The quantum Weyl group is already known to be of this form [5]. In general however, it can be hard to come up with the cocycle data required in the construction. Our first result, in Section 2, provides a general solution to the contruction problem in the case of a central extension.We give the case of U q (ŝl 2 ) in detail, in Section 3. In Section 4 we give the result more generally using the R-matrix formalism with generators l ± (z) in [6] [2].

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Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups

Šťovı́ček

1996

Journal of Algebra

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The coadjoint orbits for the series B , C , and D are considered in the case l l l when the base point is a multiple of a fundamental weight. A quantization of the big cell is suggested by means of introducing a )-algebra generated by holomorphic coordinate functions. Starting from this algebraic structure the irreducible representations of the deformed universal enveloping algebra are derived as acting in the vector space of polynomials in quantum coordinate functions.12 12 or equivalently, R y R y1 s q y q y1 P , Ž .12 21the R-matrices for the series B , C , D obey a more complicated relation l l l Ž . below.Unfortunately, this is why the construction of the algebra C C was not clarified entirely and has to some extent a speculative character. However, the prescription for the representation of U U acting in the vector space of h polynomials in quantum coordinate functions z U is derived quite unamjk biguously and verified rigorously. In principle, Section 4, concerned with the quantum parameterization of the big cell, can be skipped. But starting directly from the defining relations for the representation could seem then rather obscure.

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Quantum group Gauge theory on quantum spaces

Cited by 165 publications

References 20 publications

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

Bicrossproduct Structure of the Quantum Weyl Group

Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups

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