JournalsjncgVol. 15, No. 1pp. 305–345

Vector bundles over multipullback quantum complex projective spaces

  • Albert Jeu-Liang Sheu

    University of Kansas, Lawrence, USA
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Abstract

We work on the classification of isomorphism classes of finitely generated projective modules over the C*-algebras C(Pn(T))C(\mathbb{P}^{n}(\mathcal{T})) and C(SH2n+1)C(\mathbb{S}_{H}^{2n+1}) of the quantum complex projective spaces Pn(T)\mathbb{P}^{n}(\mathcal{T}) and the quantum spheres SH2n+1\mathbb{S}_{H}^{2n+1}, and the quantum line bundles LkL_{k} over Pn(T)\mathbb{P}^{n}(\mathcal{T}), 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(Pn(T))C(\mathbb{P}^{n}(\mathcal{T})), C(SH2n+1)C(\mathbb{S}_{H}^{2n+1}), and LkL_{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(SH2n+1)C(\mathbb{S}_{H}^{2n+1}) of rank higher than n2+3\lfloor \frac{n}{2}\rfloor+3 are free modules. Furthermore, besides identifying a large portion of the positive cone of the K0K_{0}-group of C(Pn(T))C(\mathbb{P}^{n}(\mathcal{T})), we also explicitly identify LkL_{k} with concrete representative elementary projections over C(Pn(T))C(\mathbb{P}^{n}(\mathcal{T})).

Cite this article

Albert Jeu-Liang Sheu, Vector bundles over multipullback quantum complex projective spaces. J. Noncommut. Geom. 15 (2021), no. 1, pp. 305–345

DOI 10.4171/JNCG/401