Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 015, 8 pages      arXiv:1311.4758      https://doi.org/10.3842/SIGMA.2014.015
Contribution to the Special Issue on Noncommutative Geometry and Quantum Groups in honor of Marc A. Rieffel

On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

Tomasz Brzeziński
Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK

Received December 03, 2013, in final form February 09, 2014; Published online February 14, 2014

Abstract
Recent results by Krähmer [Israel J. Math. 189 (2012), 237-266] on smoothness of Hopf-Galois extensions and by Liu [arXiv:1304.7117] on smoothness of generalized Weyl algebras are used to prove that the coordinate algebras of the noncommutative pillow orbifold [Internat. J. Math. 2 (1991), 139-166], quantum teardrops ${\mathcal O}({\mathbb W}{\mathbb P}_q(1,l))$ [Comm. Math. Phys. 316 (2012), 151-170], quantum lens spaces ${\mathcal O}(L_q(l;1,l))$ [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold ${\mathcal O}(\Sigma_q^3)$ [J. Geom. Phys. 62 (2012), 1097-1107], quantum real weighted projective planes ${\mathcal O}({\mathbb R}{\mathbb P}_q^2(l;\pm))$ [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages] and quantum Seifert lens spaces ${\mathcal O}(\Sigma_q^3(l;-))$ [Axioms 1 (2012), 201-225] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.

Key words: smooth algebra; generalized Weyl algebra; strongly graded algebra; noncommutative pillow; quantum teardrop; quantum lens space; quantum real weighted projective plane.

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