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Elmar Wagner

Publications and source records attributed to Elmar Wagner.

At least 19 recordsLinked to original sources
arXiv2025

Function algebras on the n-dimensional quantum complex space

The paper introduces a (universal) C*-algebra of continuous functions vanishing at infinity on the n-dimensional quantum complex space. To this end, the well-behaved Hilbert space representations of the defining relations are classified. Then these representations are realized by multiplication operators on an L2-space. The C*-algebra of continuous functions vanishing at infinity is defined by considering a *-algebra such that its classical counterpart separates the points of the n-dimensional complex space and by taking the operator norm closure of a universal representation of this algebra.

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arXiv2023

Closed quantum surfaces from the Toeplitz extension

Closed quantum surfaces of any genus are defined as subalgebras of the Toeplitz algebra by mimicking the classical construction of identifying arcs on the boundary of the (quantum) unit disk. Isomorphism classes obtained from different arrangements of arcs are classified. It is shown that the K-groups are isomorphic to the classical counterparts and explicit generators of the C*-algebras and of the K-groups are given.

math.QA↗
arXiv2021

A Dolbeault-Dirac Spectral Triple for the $B_2$-Irreducible Quantum Flag Manifold

The quantum version of the Bernstein-Gelfand-Gelfand resolution is used to construct a Dolbeault-Dirac operator on the anti-holomorphic forms of the Heckenberger-Kolb calculus for the $B_2$-irreducible quantum flag manifold. The spectrum and the multiplicities of the eigenvalues of the Dolbeault-Dirac operator are computed. It is shown that this construction yields an equivariant, even, $0^+$-summable spectral triple.

math.QA↗
arXiv2021

Continuous frames in Krein spaces

The purpose of this paper is to propose a definition of continuous frames of rank n for Krein spaces and to study their basic properties. Similarly to the Hilbert space case, continuous frames are characterized by the analysis, the pre-frame and the frame operator, where the latter gives rise to a frame decomposition theorem. The paper includes a discussion of similar, dual and Parseval frames and of reproducing kernels. In addition, the importance of the fundamental symmetry in the formula for the frame operator in a Krein space is clarified. As prime examples, it is shown how to transfer continuous frames for Hilbert spaces to Krein spaces arising from a possibly non-regular Gram operator.

math.FA↗
arXiv2020

Duality for infinite-dimensional braided bialgebras and their (co)modules

The paper presents a detailed description of duality for braided algebras, coalgebras, bialgebras, Hopf algebras and their modules and comodules in the infinite setting. Assuming that the dual objects exist, it is shown how a given braiding induces compatible braidings for the dual objects, and how actions (resp. coactions) can be turned into coactions (resp. actions) of the dual coalgebra (resp. algebra), with an emphasis on braided bialgebras and their braided (co)module algebras.

math.QA↗
arXiv2020

A spectral triple for noncommutative compact surfaces

A Dirac operator is presented that will yield a 1+ summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes' conditions for noncommutative spin geometries are analyzed and it is argued that the failure of some requirements is mainly due to a wrong choice of a noncommutative spin bundle.

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arXiv2019

Twisted Dirac operator on quantum SU(2) in disc coordinates

The quantum disc is used to define a noncommutative analogue of a dense coordinate chart and of left-invariant vector fields on quantum SU(2). This yields two twisted Dirac operators for different twists that are related by a gauge transformation and have bounded twisted commutators with a suitable algebra of differentiable functions on quantum SU(2).

math.QA↗
arXiv2018

K-theory and index pairings for C*-algebras generated by q-normal operators

The paper presents a detailed description of the K-theory and K-homology of C*-algebras generated by q-normal operators including generators and the index pairing. The C*-algebras generated by q-normal operators can be viewed as a q-deformation of the quantum complex plane. In this sense, we find deformations of the classical Bott projections describing complex line bundles over the 2-sphere, but there are also simpler generators for the K_0-groups, for instance 1-dimensional Powers-Rieffel type projections and elementary projections belonging to the C*-algebra. The index pairing between these projections and generators of the even K-homology group is computed, and the result is used to express the K_0-classes of the quantized line bundles of any winding number in terms of the other projections.

math.QA↗
arXiv2018

Toeplitz algebras in quantum Hopf fibrations

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 Podles spheres which are isomorphic to those from the well-known 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 Podles 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.

math.QA↗
arXiv2018

Function algebras on a 2-dimensional quantum complex plane

The well-behaved representations of the coordinate algebra of a 2-dimensional quantum complex plane are classified and a C*-algebra is defined which can be viewed as the algebra of continuous functions on the 2-dimensional quantum complex plane vanishing at infinity.

math.QA↗
arXiv2018

Representations of quantum SU(2) operators on a local chart

Hilbert space representations of quantum SU(2) by multiplication operators on a local chart are constructed, where the local chart is given by tensor products of square integrable functions on a quantum disc and on the classical unit circle. The actions of generators of quantum SU(2), generators of the opposite algebra, and noncommutative partial derivatives are computed on a Hilbert space basis.

math.QA↗
arXiv2018

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

It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podles spheres are related by restriction. In this approach, the equatorial Podles 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.

math.QA↗
arXiv2018

A noncommutative 2-sphere generated by the quantum complex plane

S. L. Woronowicz's theory of introducing 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.

math.QA↗
arXiv2014

Braided join comodule algebras of Galois objects

We construct the join of noncommutative Galois objects (quantum torsors) over a Hopf algebra H. To ensure that the join algebra enjoys the natural (diagonal) coaction of H, we braid the tensor product of the Galois objects. Then we show that this coaction is principal. Our examples are built from the noncommutative torus with the natural free action of the classical torus, and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras. The former yields a noncommutative deformation of a non-trivial torus bundle, and the latter a finite quantum covering.

math.QA↗
arXiv2013

Frames in Krein spaces arising from a non-regular W-metric

A definition of frames in Krein spaces is stated and a complete characterization is given by comparing them to frames in the associated Hilbert space. The basic tools of frame theory are described in the formalism of Krein spaces. It is shown how to transfer a frame for Hilbert spaces to Krein spaces given by a W-metric, where the Gram operator W is not necessarily regular and possibly unbounded.

math.FA↗
arXiv2013

The Pullbacks of Principal Coactions

We prove that the class of principal coactions is closed under one-surjective pullbacks in an appropriate category of algebras equipped with left and right coactions. This allows us to handle cases of C*-algebras lacking two different non-trivial ideals. It also allows us to go beyond the category of comodule algebras. As an example of the former, we carry out an index computation for noncommutative line bundles over the standard Podles sphere using the Mayer-Vietoris type arguments afforded by a one-surjective pullback presentation of the C*-algebra of this quantum sphere. To instantiate the latter, we define a family of coalgebraic noncommutative deformations of the U(1)-principal bundle S^7 --> CP^3.

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