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1
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Restricting the bi-equivariant spectral triple on quantum SU(2) to the Podleś spheres

100%
Wagner E.
Banach Center Publications
|
2011
|
tom 93
|
nr 1
225-240
EN
It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podleś spheres are related by restriction. In this approach, the equatorial Podleś sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is shown that the failure of the real structure to satisfy the commutant property is related to the failure of the universal R-matrix operator to be unitary.
2
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A noncommutative 2-sphere generated by the quantum complex plane

63%
Cohen I., Wagner E.
Banach Center Publications
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2012
|
tom 98
|
nr 1
55-66
EN
S. L. Woronowicz's theory of C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.
3
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Twisted spectral triples and covariant differential calculi

63%
Krähmer U., Wagner E.
Banach Center Publications
|
2011
|
tom 93
|
nr 1
177-188
EN
Connes and Moscovici recently studied "twisted" spectral triples (A,H,D) in which the commutators [D,a] are replaced by D∘a - σ(a)∘D, where σ is a second representation of A on H. The aim of this note is to point out that this yields representations of arbitrary covariant differential calculi over Hopf algebras in the sense of Woronowicz. For compact quantum groups, H can be completed to a Hilbert space and the calculus is given by bounded operators. At the end, we discuss an explicit example of Heckenberger's 3-dimensional covariant differential calculi on quantum SU(2).
4
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Index pairings for pullbacks of C*-algebras

45%
Dąbrowski L., Hadfield T., Hajac P., Matthes R., Wagner E.
Banach Center Publications
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2012
|
tom 98
|
nr 1
67-84
EN
In this overview, we study how to reduce the index pairing for a fibre-product C*-algebra to the index pairing for the C*-algebra over which the fibre product is taken. As an example we analyze the case of suspensions and apply it to noncommutative instanton bundles of arbitrary charges over the suspension of quantum deformations of the 3-sphere.
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