Classification of three-dimensional covariant differential calculi on Podles' quantum spheres and on related spaces

Apel, Joachim; Schmüdgen, Konrad

doi:10.1007/bf00761121

Cited by 16 publications

(21 citation statements)

References 8 publications

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“…For the application given in Section 6 it is crucial that this calculus is induced from the 3D-calculus of the quantum group SU q (2). This fact has been known to the author since several years (in fact, since the writing of [1]) and also to others (S. Majid, P. Podles). Since I could not find this result in the literature, we shall derive it in this subsection.…”

Section: The 2-dimensional Calculus On the Quantum Sphere S 2 Qmentioning

confidence: 65%

“…It is well-known that that O(S 2 q ) is the coordinate algebra of the Podles' quantum 2-sphere S 2 qc in the case c = 0 [9]. (Note that the quantum 2-spheres in [9], [10] and [1] are right quantum spaces, while we consider the corresponding left quantum spaces here.) The generators x + , x − , y 0 satisfy the relations…”

Section: The 2-dimensional Calculus On the Quantum Sphere S 2 Qmentioning

confidence: 99%

“…We shall not need this in this paper, because we consider the coordinate * -algebra O(SU q (2)) which has only bounded * -representations.) The above notion is closely related to Alain Connes' concept of a K-cycle 1 . If in addition F is a self-adjoint operator with compact resolvent and the commutator [π(x), F ] is bounded for any x ∈ A the pair (π, F ) is called a K-cycle over the * -algebra A.…”

Section: Introductionmentioning

confidence: 97%

“…Therefore, it is sufficient to prove that ρ(xω Γ1 (by)) = 0 for all b ∈ B and x, y ∈ A. Indeed, using the facts that ρ is a bimodule homomorphism and ω Γ1 (by) = S(y (1) )ω Γ1 (b)y (2) (see formula (14.3) in [6]) we obtain…”

Section: Introductionmentioning

confidence: 99%

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Commutator representations of differential calculi on the quantum group SUq(2)

Schmüdgen

1999

Journal of Geometry and Physics

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Let (Γ, d) be the 3D-calculus or the 4D±-calculus on the quantum group SUq(2). We describe all pairs (π, F ) of a * -representation π of O(SUq (2)) and of a symmetric operator F on the representation space satisfying a technical condition concerning its domain such that there exist a homomorphism of first order differential calculi which maps dx into the commutator [iF, π(x)] for x ∈ O(SUq(2)). As an application commmutator representations of the 2-dimensional leftcovariant calculus on Podles quantum 2-sphere S 2 qc with c = 0 are given.

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Section: The 2-dimensional Calculus On the Quantum Sphere S 2 Qmentioning

confidence: 65%

Section: The 2-dimensional Calculus On the Quantum Sphere S 2 Qmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Commutator representations of differential calculi on the quantum group SUq(2)

Schmüdgen

1999

Journal of Geometry and Physics

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“…The equivariant derivations of B induced from those of A and e.g. the calculi in [1], [4], [10], [11] also correspond to right ideals of B + in this way. For B = A, Woronowicz [16] shows that all equivariant derivations have this property.…”

Section: And (C) (A) This Proves Assertion (Iii)mentioning

confidence: 99%

Derivations With Quantum Group Action

Hermisson

2002

Communications in Algebra

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Abstract. The derivations of a left coideal subalgebra B of a Hopf algebra A which are compatible with the comultiplication of A (that is, the covariant first order differential calculi, as defined by Woronowicz, on a quantum homogeneous space) are related to certain right ideals of B. The correspondence is one-to-one if A is faithfully flat as a right B-module. This generalizes the result for B = A due to Woronowicz. A definition for the dimension of a first order differential calculus at a classical point is given. For the quantum 2-sphere S 2 qc of Podleś under the assumptions q n+1 = 1 and c = −q 2n /(q 2n + 1) 2 for all n = 0, 1, . . . , three 2-dimensional covariant first order differential calculi exist if c = 0, one exists if c = ∓q/(±q + 1) 2 and none else. This extends a result of Podleś. PRELIMINARIESA derivation of an algebra B over C (the complex numbers) is defined as a C-linear map d from B into a B-bimodule satisfying the Leibniz ruleIn this paper, da b means (da) b. We set Γ(d) = Lin C {a db | a, b ∈ B} (the C-linear span). We write d ′ ≤ d, if d ′ and d are derivations of B and

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