Among the popular and well analyzed quantum spaces $X_{q}$ with their $K$-groups $K^{i}\left( X_{q}\right) \equiv K_{i}\left( C\left(X_{q}\right) \right) $ already computed, are the multipullback quantum odd-dimensional spheres $\mathbb{S}_{H}^{2n+1}$ and the associated quantum complex projective spaces $\mathbb{P}^{n}\left( \mathcal{T}\right) $, introduced and studied by Hajac, Kaygun, Nest, Pask, Sims, and Zielinski. In noncommutative geometry, finitely generated projective modules (f.g.p.m.) over $C\left( X_{q}\right)$, efficiently encoded by projections over $C\left( X_{q}\right) $, are viewed as (quantum) vector bundles over $X_{q}%$, and are classified up to stable isomorphism by the positive cone of $K_{0}\left( C\left( X_{q}\right) \right) $. With the $K_{0}$-groups of $C\left( \mathbb{S}_{H}^{2n+1}\right) $ and $C\left( \mathbb{P}^{n}\left(\mathcal{T}\right) \right) $ known, it is natural to seek the classification of vector bundles over $\mathbb{S}_{H}^{2n+1}$ and $\mathbb{P}^{n}\left(\mathcal{T}\right) $ up to isomorphism. Following an approach popularized by Curto, Muhly, and Renault, we first realize $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $ and $C\left( \mathbb{S}_{H}^{2n+1}\right) $ as groupoid $C^*$-algebras to better understand their structures in the framework of groupoid $C^*$-algebras, which also provides a convenient context for discussing projections over them. Then we apply Rieffel's theory of stable ranks to derive some answers regarding the classification of f.g.p.m. over the $n$-dimensional Toeplitz algebra $\mathcal{T}^{\otimes n}$, $C\left( \mathbb{S}_{H}^{2n+1}\right) $, and $C\left( \mathbb{P}^{n}\left( \mathcal{T}\right) \right) $. In particular, as concrete direct sums of elementary projections over $C\left(\mathbb{P}^{n}\left( \mathcal{T}\right) \right) $, we can identify those distinguished quantum line bundles $L_{k}$ over $\mathbb{P}^{n}\left(\mathcal{T}\right) $ for $k\in\mathbb{Z}$ that were constructed from quantum principal $U\left( 1\right) $-bundles by Hajac and his collaborators,rendering their module structures transparent.
You are here
Line Bundles over Multipullback Quantum Complex Projective Spaces
Research Group:
Speaker:
Albert Jeu-Liang Sheu
Institution:
University of Kansas
Schedule:
Wednesday, April 11, 2018 - 11:30 to 12:30
Location:
A-133
Abstract:
Openings
- Call of interest for positions in mathematics at SISSA
- Public Calls for Professors
- Temporary Researchers
- Post Doctoral Fellowships
- Open positions in MathLab
- SISSA Mathematical Fellowships
- PhD Scolarships
- SIS Fellowships
- Pre-PhD Fellowships
- MSc in Mathematics
- Master Degree in Data Science and Scientific Computing
- Professional Master Courses
- SISSA Mathematics Medals
Upcoming events
-
Valentina Beorchia, Valeria Chiadò Piat, Maria Strazullo
SISSA Women in Mathematics 2024
Monday, May 13, 2024 - 14:15
-
Antonio De Rosa
Min-max construction of anisotropic CMC surfaces
Thursday, May 16, 2024 - 14:00 to 16:00
-
Mattia Fogagnolo
TBA
Thursday, May 30, 2024 - 14:00 to 16:00
-
Marco Pozzetta
TBA
Thursday, June 13, 2024 - 14:00 to 15:00
Today's Lectures
-
10:15 to 13:00
-
14:00 to 17:00
-
14:15 to 16:00
-
16:30 to 17:15
Recent publications
-
G.P. Leonardi; G. Saracco,Rigidity and trace properties...
-
R. Marchello; A. Colombi; L. Preziosi; C. Giverso,A non local model for cell mig...
-
I. Prusak; D. Torlo; M. Nonino; G. Rozza,Optimisation–Based Coupling of...
-
A.Surya Boiardi; R. Marchello,Breaking the left-right symmet...