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elke

### Albert Sheu (University of Kansas): Line Bundles over Multipullback Quantum Complex Projective Spaces. Oberseminar C*-Algebren

##### am Dienstag, 24.04.2018 15:15 im Raum N2

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 Zieli\'{n}ski. 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.

Angelegt am Donnerstag, 18.01.2018 15:01 von elke
Geändert am Freitag, 09.03.2018 09:12 von elke
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