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Albert Jeu-Liang Sheu

Publications and source records attributed to Albert Jeu-Liang Sheu.

9 recordsLinked to original sources
arXiv2020

Nonstandard Quantum Complex Projective Line

In our attempt to explore how the quantum nonstandard complex projective spaces $\mathbb{C}P_{q,c}^{n}$ studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of $C\big(\mathbb{C}P_{q,c}^{1}\big) $ with the pull-back of two copies of the Toeplitz $C^*$-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting non-obvious details, such as a prominent generator of $C\big(\mathbb{C}P_{q,c}^{1}\big) $ being a concrete weighted double shift.

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arXiv2019

Projections over Quantum Homogeneous Odd-dimensional Spheres

We give a complete classification of isomorphism classes of finitely generated projective modules, or equivalently, unitary equivalence classes of projections, over the C*-algebra $C\left( \mathbb{S}_{q}^{2n+1}\right) $ of the quantum homogeneous sphere $\mathbb{S}_{q}^{2n+1}$. Then we explicitly identify as concrete elementary projections the quantum line bundles $L_{k}$ over the quantum complex projective space $\mathbb{C}P_{q}^{n}$ associated with the quantum Hopf principal $U\left( 1\right) $-bundle $\mathbb{S} _{q}^{2n+1}\rightarrow\mathbb{C}P_{q}^{n}$.

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arXiv2018

Vector Bundles over Multipullback Quantum Complex Projective Spaces

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T} \right) $ and the quantum spheres $\mathbb{S}_{H}^{2n+1}$, and the quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left( \mathcal{T}\right) $, studied by Hajac and collaborators. Motivated by the groupoid approach of Curto, Muhly, and Renault to the study of C*-algebraic structure, we analyze $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, $C\left( \mathbb{S}_{H}^{2n+1}\right) $, and $L_{k}$ in the context of groupoid C*-algebras, and then apply Rieffel's stable rank results to show that all finitely generated projective modules over $C\left( \mathbb{S}_{H} ^{2n+1}\right) $ of rank higher than $\left\lfloor \frac{n}{2}\right\rfloor +3$ are free modules. Furthermore, besides identifying a large portion of the positive cone of the $K_{0}$-group of $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, we also explicitly identify $L_{k}$ with concrete representative elementary projections over $C\left( \mathbb{P} ^{n}\left( \mathcal{T}\right) \right) $.

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arXiv2018

A Note on the Quantum Family of Maps

The notion and theory of the quantum space of all maps from a quantum space pioneered by Sołtan have been mainly focused on finite-dimensional C*-algebras which are matrix algebra bundles over a finite set $S$. We propose a modification of this notion to cover the case of $C\left( X\right) $ for general compact Hausdorff spaces $X$ instead of finite sets $S$ while taking into account of the topology of $X$. A notion of free product of copies of a unital C*-algebra topologically indexed by a compact Hausdorff space arises naturally, and satisfies some desired functoriality.

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arXiv2018

Projective Modules over Quantum Projective Line

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right) $ constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right) $ realized as a concrete groupoid C*-algebra, and find its $K$-groups. Furthermore after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra $C\left( \mathbb{P}^{1}\left( \mathcal{T}\right) \right) $, we identify those quantum principal $U\left( 1\right) $-bundles introduced by Hajac and collaborators among the projections classified.

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arXiv2014

The Structure of Line Bundles over Quantum Teardrops

Over the quantum weighted 1-dimensional complex projective spaces, called quantum teardrops, the quantum line bundles associated with the quantum principal U(1)-bundles introduced and studied by Brzezinski and Fairfax are explicitly identified among the finitely generated projective modules which are classified up to isomorphism. The quantum lens space in which these quantum line bundles are embedded is realized as a concrete groupoid C*-algebra.

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arXiv1998

Quantum Spheres As Groupoid C*-algebras

We show that the C*-algebra of a quantum sphere $C(S_{q}^{2n+1})$ can be realized as a groupoid C*-algebra of a groupoid which is explicitly identified and is independent of the parameter $q$.

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