On the Smoothness of the Noncommutative Pillow and Quantum Teardrops

2013/11/30 by Tomasz Brzeziński
Mathematics · #math.QA #msc:58B32 #msc:58B34

paper · pdf · doi:10.3842/sigma.2014.015

published as SIGMA 10 (2014), 015, 8 pages

arxiv created 2014/02/14 · arxiv updated 2014/02/17

Abstract

Recent results by Krähmer [Israel J. Math. 189 (2012), 237-266, arXiv:0806.0267] 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, arXiv:1107.1417], quantum lens spaces \mathcal O(L_q(l;1,l)) [Pacific J. Math. 211 (2003), 249-263], the quantum Seifert manifold \mathcal O(Σ_q³) [J. Geom. Phys. 62 (2012), 1097-1107, arXiv:1105.5897], quantum real weighted projective planes \mathcal O(\mathbb R\mathbb P_q²(l;±)) [PoS Proc. Sci. (2012), PoS(CORFU2011), 055, 10 pages, arXiv:1203.6801] and quantum Seifert lens spaces \mathcal O(Σ_q³(l;-)) [Axioms 1 (2012), 201-225, arXiv:1207.2313] are homologically smooth in the sense that as their own bimodules they admit finitely generated projective resolutions of finite length.

Citations

• On the Hochschild (co)homology of Quantum Homogeneous Spaces
• Homological smoothness and deformations of generalized Weyl algebras
• Circle actions on a quantum Seifert manifold
• Bundles over Quantum Real Weighted Projective Spaces