Abstract
The quantum weighted projective algebras \(\mathbb {C}[\mathbb {W}\mathbb {P}_{k,l,q}]\) are coinvariant subalgebras of the quantum group algebra \(\mathbb {C}[SU_{2,q}]\). For each pair of indices k,l, two 2-summable spectral triples will be constructed. The first one is an odd spectral triple based on coinvariant spinors on \(\mathbb {C}[SU_{2,q}]\). The second one is an even spectral triple.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brzezinski, T., Fairfax, S.A.: Quantum Teardrops. Commun. Math. Phys. 316, 151–170 (2012)
Dabrowski, L., Landi, G., Sitarz, A., van Suijlekom, W., Varilly, J.C.: The Dirac Operator on S U q (2). Commun. Math. Phys. 259, 729–759 (2005)
Drinfeld, V.G.: Quantum groups. In: Gleason A. M. (ed.) Procedings I.C.M. Berkeley (1986)
Gracia-Bondia, J.M., Varilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhauser (2001)
Harju, A.J.: Spectral Triples on Proper Etale Groupoids. to appear in J. Noncom. Geom. (2014). preprint: arXiv
Jimbo, M.: A q-Difference Analogue of U(g) and the Yang-Baxter Equation. Lett. Math. Phys. 10, 63–69 (1985)
Klimyk, A., Schmudgen, K.: Quantum groups and their representations. Springer (1998)
Moerdijk, I.: Orbifolds as groupoids: an introduction. Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math. 310, 205-222 (Amer. Math. Soc., Providence), RI (2002)
Moerdijk, I., Mrcun, J.: Introduction to Foliations and Lie Groupoids. Cambridge University Press (2003)
Neshveyev, S., Tuset, L.: The Dirac operator on compact quantum groups. J. Reine. Angew. Math 641, 1–20 (2010)
Neshveyev, S., Tuset, L.: Notes on the KazhdanLusztig Theorem on Equivalence of the Drinfeld Category and the Category of \(U_{q} \mathfrak {g}\)-Modules. Algebr. Represent. Theory 14, 897–948 (2011)
Rennie, A.C., Varilly, J.C.: Orbifolds are not commutative geometries. J. Australian Math. Soc. 84, 109–116 (2008)
Satake, I.: On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. U.S.A. 42, 359–363 (1956)
Sitarz, A., Venselaar, J.J.: Real spectral triples on 3-dimensional noncommutative lens spaces. preprint: arXiv (2013)
Sheu, A.J.L.: The Structure of Line Bundles over Quantum Teardrops. SIGMA 10 (2014)
Slebarski, S.: Dirac operators on a compact Lie group. Bull. London Math. Soc. 17, 579–583 (1985)
Woronowicz, S.L.: Twisted SU(2) Group. An Example of a Non-Commutative Differential Calculus. Publ. Res. Inst. Math. Sci 23, 117–181 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Stanislaw Lech Woronowicz.
Rights and permissions
About this article
Cite this article
Harju, A.J. Dirac Operators on Quantum Weighted Projective Spaces. Algebr Represent Theor 18, 1187–1210 (2015). https://doi.org/10.1007/s10468-015-9536-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-015-9536-9