Albert Sheu (University of Kansas): Line Bundles over Multipullback Quantum Complex Projective Spaces. Oberseminar C*-Algebren
Tuesday, 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 18.01.2018 von Elke Enning
Geändert am 09.03.2018 von Elke Enning
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