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A noncommutative 2-sphere generated by the quantum complex plane

Ismael Cohen, Elmar Wagner (2012)

Banach Center Publications

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...

Covariantization of quantized calculi over quantum groups

Seyed Ebrahim Akrami, Shervin Farzi (2020)

Mathematica Bohemica

We introduce a method for construction of a covariant differential calculus over a Hopf algebra A from a quantized calculus d a = [ D , a ] , a ∈ A , where D is a candidate for a Dirac operator for A . We recover the method of construction of a bicovariant differential calculus given by T. Brzeziński and S. Majid created from a central element of the dual Hopf algebra A ∘ . We apply this method to the Dirac operator for the quantum SL ( 2 ) given by S. Majid. We find that the differential calculus obtained by our method is the...

Differential calculi over quantum groups

G. J. Murphy (2003)

Banach Center Publications

An exposition is given of recent work of the author and others on the differential calculi that occur in the setting of compact quantum groups. The principal topics covered are twisted graded traces, an extension of Connes' cyclic cohomology, invariant linear functionals on covariant calculi and the Hodge, Dirac and Laplace operators in this setting. Some new results extending the classical de Rham theorem and Poincaré duality are also discussed.

Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two

Tomasz Brzeziński (2015)

Colloquium Mathematicae

Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

Finite closed coverings of compact quantum spaces

Piotr M. Hajac, Atabey Kaygun, Bartosz Zieliński (2012)

Banach Center Publications

We consider the poset of all non-empty finite subsets of the set of natural numbers, use the poset structure to topologise it with the Alexandrov topology, and call the thus obtained topological space ℙ ∞ the universal partition space. Then we show that it is a classifying space for finite closed coverings of compact quantum spaces in the sense that any such a covering is functorially equivalent to a sheaf over this partition space. In technical terms, we prove that the category of finitely supported...

Localizations for construction of quantum coset spaces

Zoran Škoda (2003)

Banach Center Publications

Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum...

Noncommutative Borsuk-Ulam-type conjectures

Paul F. Baum, Ludwik Dąbrowski, Piotr M. Hajac (2015)

Banach Center Publications

Within the framework of free actions of compact quantum groups on unital C*-algebras, we propose two conjectures. The first one states that, if δ : A → A ⨂ m i n H is a free coaction of the C*-algebra H of a non-trivial compact quantum group on a unital C*-algebra A, then there is no H-equivariant *-homomorphism from A to the equivariant join C*-algebra A ⊛ δ H . For A being the C*-algebra of continuous functions on a sphere with the antipodal coaction of the C*-algebra of functions on ℤ/2ℤ, we recover the celebrated Borsuk-Ulam...

Quantum 4-sphere: the infinitesimal approach

F. Bonechi, M. Tarlini, N. Ciccoli (2003)

Banach Center Publications

We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to q ( S U ( 2 ) ) .

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