Piecewise principal comodule algebras

Hajac, Piotr M.; Krähmer, Ulrich; Matthes, Rainer; Zieliński, Bartosz

doi:10.4171/jncg/88

Cited by 42 publications

(116 citation statements)

References 30 publications

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“…However, even in this special case of the equivariant join comodule algebra, our non-unital strongconnection formula is differently constructed than the complicated strong-connection formula in [7,Lemma 3.2]. This might be very important for index pairing computations involving concrete strong-connection formulas.…”

Section: Definition 32 ([3])mentioning

confidence: 99%

“…Observe that in the special case of the join construction, we can take t to be the inclusion map [0, 1] → C. Moreover, as explained in the next section, the equivariant join comodule algebra P δ H becomes piecewise trivial, and Theorem 5.3 follows from [7,Lemma 3.2]. However, even in this special case of the equivariant join comodule algebra, our non-unital strongconnection formula is differently constructed than the complicated strong-connection formula in [7,Lemma 3.2].…”

Section: Definition 32 ([3])mentioning

confidence: 99%

“…This section is an adjustment of [7,Section 5.1] to the setting of equivariant join construction. It is important to bear in mind that we cannot expect to have such a piecewise (pullback) structure in the general case of equivariant fusion comodule algebra.…”

Section: Piecewise Structurementioning

confidence: 99%

“…To make this paper self-contained and to establish notation and terminology, we begin by recalling the basics of classical joins, strong connections [6] and principal comodule algebras [7]. In the first section, we define the join and fusion C * -algebra of an arbitrary pair of unital C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

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Equivariant Join and Fusion of Noncommutative Algebras

Da̧browski

Hadfield²,

Hajac³

2015

SIGMA

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Abstract. We translate the concept of the join of topological spaces to the language of C * -algebras, replace the C * -algebra of functions on the interval [0, 1] with evaluation maps at 0 and 1 by a unital C * -algebra C with appropriate two surjections, and introduce the notion of the fusion of unital C * -algebras. An appropriate modification of this construction yields the fusion comodule algebra of a comodule algebra P with the coacting Hopf algebra H. We prove that, if the comodule algebra P is principal, then so is the fusion comodule algebra. When C = C([0, 1]) and the two surjections are evaluation maps at 0 and 1, this result is a noncommutative-algebraic incarnation of the fact that, for a compact Hausdorff principal G-bundle X, the diagonal action of G on the join X * G is free.

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Section: Definition 32 ([3])mentioning

confidence: 99%

Section: Definition 32 ([3])mentioning

confidence: 99%

Section: Piecewise Structurementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivariant Join and Fusion of Noncommutative Algebras

Da̧browski

Hadfield²,

Hajac³

2015

SIGMA

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“…In algebraic terms, quantum principal bundles are given by principal comodule algebras; see e.g. [13]. These are examples of Hopf-Galois extensions whose definition and rudimentary properties we recall presently.…”

Section: 2mentioning

confidence: 99%

Non-commutative integral forms and twisted multi-derivations

Brzeziński¹,

Kaoutit²,

Lomp³

2010

J. Noncommut. Geom.

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Abstract. Non-commutative connections of the second type or hom-connections and associated integral forms are studied as generalisations of right connections of Manin. First, it is proven that the existence of hom-connections with respect to the universal differential graded algebra is tantamount to the injectivity, and that every injective module admits a homconnection with respect to any differential graded algebra. The bulk of the article is devoted to describing a method of constructing hom-connections from twisted multi-derivations. The notion of a free twisted multi-derivation is introduced and the induced first order differential calculus is described. It is shown that any free twisted multi-derivation on an algebra A induces a unique hom-connection on A (with respect to the induced differential calculus 1 .A/) that vanishes on the dual basis of 1 .A/. To any flat hom-connection r on A one associates a chain complex, termed a complex of integral forms on A. The canonical cokernel morphism to the zeroth homology space is called a r-integral. Examples of free twisted multi-derivations, hom-connections and corresponding integral forms are provided by covariant calculi on Hopf algebras (quantum groups). The example of a flat hom-connection within the 3D left-covariant differential calculus on the quantum group O q .SL.2// is described in full detail. A descent of hom-connections to the base algebra of a faithfully flat Hopf-Galois extension or a principal comodule algebra is studied. As an example, a hom-connection on the standard quantum Podleś sphere O q .S 2 / is presented. In both cases the complex of integral forms is shown to be isomorphic to the de Rham complex, and the r-integrals coincide with Hopf-theoretic integrals or invariant (Haar) measures.Mathematics Subject Classification (2010). 58B32; 16W25.

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Toeplitz Algebras in Quantum Hopf Fibrations

Wagner

2012

Recent Progress in Operator Theory and Its Applications

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The paper presents applications of Toeplitz algebras in Noncommutative Geometry. As an example, a quantum Hopf fibration is given by gluing trivial U(1) bundles over quantum discs (or, synonymously, Toeplitz algebras) along their boundaries. The construction yields associated quantum line bundles over the generic Podleś spheres which are isomorphic to those from the wellknown Hopf fibration of quantum SU(2). The relation between these two versions of quantum Hopf fibrations is made precise by giving an isomorphism in the category of right U(1)-comodules and left modules over the C*-algebra of the generic Podleś spheres. It is argued that the gluing construction yields a significant simplification of index computations by obtaining elementary projections as representatives of K-theory classes. * MSC2010: 47L80; 81R50

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Piecewise principal comodule algebras

Cited by 42 publications

References 30 publications

Equivariant Join and Fusion of Noncommutative Algebras

Equivariant Join and Fusion of Noncommutative Algebras

Non-commutative integral forms and twisted multi-derivations

Toeplitz Algebras in Quantum Hopf Fibrations

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