BICOVARIANT CALCULUS ON TWISTED <font>ISO</font>(N), QUANTUM POINCARÉ GROUP AND QUANTUM MINKOWSKI SPACE

Aschieri, Paolo; Castellani, Leonardo

doi:10.1142/s0217751x96002091

Cited by 30 publications

(46 citation statements)

References 17 publications

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“…A similar argument as above then shows that all ω (n) have entries that are central (as one forms) in Ω(S 7 θ ′ ); again, this means that L θ (ω (n) ) = ω (n) . In particular, this holds for the adjoint bundle associated to the adjoint representation on su(2) C ≃ C 3 (as complex representation spaces), from which we conclude that ∇ (2) 0 coincides with [∇ 0 , ·] (since this is the case if θ = 0).…”

Section: Lemma 18 There Is the Following Isomorphism Of Rightmentioning

confidence: 56%

“…In the undeformed case, the infinitesimal gauge potentials generated by acting with elements in so(5, 1) − so(5) on the basic instanton gauge potential ω (0) satisfy (∇ (2) 0 ) * (δ i ω (0) ) = 0 as shown in [6]. The result then follows from the observation that ∇ (2) 0 commutes with the quantization map L θ (cf. Remark 19).…”

Section: Twisted Conformal Transformationsmentioning

confidence: 88%

“…Their use to implement symmetries of toric noncommutative manifolds like the ones of the present paper was made explicit in [35]. The geometry of multi-parametric quantum groups and quantum enveloping algebras coming from twists has been studied in [2,3].…”

Section: Twisted Infinitesimal Symmetriesmentioning

confidence: 99%

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Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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We construct a five-parameter family of gauge-nonequivalent SU (2) instantons on a noncommutative four sphere S 4 θ and of topological charge equal to 1. These instantons are critical points of a gauge functional and satisfy self-duality equations with respect to a Hodge star operator on forms on S 4 θ . They are obtained by acting with a twisted conformal symmetry on a basic instanton canonically associated with a noncommutative instanton bundle on the sphere. A completeness argument for this family is obtained by means of index theorems. The dimension of the "tangent space" to the moduli space is computed as the index of a twisted Dirac operator and turns out to be equal to five, a number that survives deformation.

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Section: Lemma 18 There Is the Following Isomorphism Of Rightmentioning

confidence: 56%

Section: Twisted Conformal Transformationsmentioning

confidence: 88%

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Noncommutative Instantons from Twisted Conformal Symmetries

Landi

Suijlekom

2007

Commun. Math. Phys.

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“…Let us introduce U op q so(N ) the Hopf algebra with the same algebra structure of U q so(N ) , but opposite coalgebra, and by ∆ op (g) = g (2) ⊗ g (1) its coproduct. On any module algebra M of U op q so(N ) the corresponding action op ⊳ will thus fulfill the relations…”

Section: Inner Derivations and Framementioning

confidence: 99%

Geometrical Tools for Quantum Euclidean Spaces

Cerchiai

Fiore

Madore

2001

Commun. Math. Phys.

106

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We apply one of the formalisms of noncommutative geometry to R N q , the quantum space covariant under the quantum group SO q (N ). Over R N q there are two SO q (N )-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of R 3 q . As in the case N = 3, one has to slightly enlarge the algebra R N q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N = 3 there is a conformal ambiguity in the natural metrics on the differential calculi over R N q . While in our previous article the frame was found 'by hand', here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of R N q with U q so(N ) into R N q , an interesting result in itself.

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“…These graded-involutive differential algebras turn out to be graded-involutive differential Hopf algebras (with coproducts and counits extending the original ones) which, in view of [5], means that the corresponding differential calculi are bicovariant in the sense of [57]. It is worth noticing that the above θ-deformations of R m , of the differential calculus on R m and of some classical groups have been already considered for instance in [3]. The quantum group setting analysis of [3] is clearly very interesting: There, R m θ appears (with other notations) as a quantum space on which some quantum group acts (or more precisely as a quantum homogeneous space) and the differential calculus on R m θ is the covariant one.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Connes¹,

Dubois-Violette²

2002

Commun Math Phys

169

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We exhibit large classes of examples of noncommutative finite-dimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations S 3 u of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4 . For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. It follows that different S 3 u can span the same R 4 u . This equivalence generates a foliation of the parameter space Σ. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two 1-parameter families C± ⊂ Σ. Up to the simple operation of taking the fixed algebra by an involution, these two families are identical and we concentrate here on C+. For u ∈ C+ the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θ-deformations, a notion which we generalize in any dimension and various contexts, and study in some details. Here, and this point is crucial, the dimension is not an artifact, i.e. the dimension of the classical model, but is the Hochschild dimension of the corresponding algebra which remains constant during the deformation. Besides the standard noncommutative tori, examples of θ-deformations include the recently defined noncommutative 4-sphere S 4 θ as well as m-dimensional generalizations, noncommutative versions of spaces R m and quantum groups which are deformations of various classical groups. We develop general tools such as the twisting of the Clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative 2n-dimensional spherical manifolds S 2n θ . A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.

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BICOVARIANT CALCULUS ON TWISTED ISO(N), QUANTUM POINCARÉ GROUP AND QUANTUM MINKOWSKI SPACE

Cited by 30 publications

References 17 publications

Noncommutative Instantons from Twisted Conformal Symmetries

Noncommutative Instantons from Twisted Conformal Symmetries

Geometrical Tools for Quantum Euclidean Spaces

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