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

SIGMA 15 (2019), 043, 8 pages      arXiv:1904.00272      https://doi.org/10.3842/SIGMA.2019.043

A Note on Spectral Triples on the Quantum Disk

Slawomir Klimek ^a, Matt McBride ^b and John Wilson Peoples ^a
a) Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202, USA
^b) Department of Mathematics and Statistics, Mississippi State University, 175 President's Cir., Mississippi State, MS 39762, USA

Received April 03, 2019, in final form May 24, 2019; Published online May 28, 2019

Abstract
By modifying the ideas from our previous paper [SIGMA 13 (2017), 075, 26 pages, arXiv:1705.04005], we construct spectral triples from implementations of covariant derivations on the quantum disk.

Key words: invariant and covariant derivations; spectral triple; quantum disk.

pdf (283 kb)   tex (12 kb)  

References

1. Carey A.L., Klimek S., Wojciechowski K.P., A Dirac type operator on the non-commutative disk, Lett. Math. Phys. 93 (2010), 107-125.
2. Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994.
3. Connes A., Cyclic cohomology, quantum group symmetries and the local index formula for ${\rm SU}_q(2)$, J. Inst. Math. Jussieu 3 (2004), 17-68, arXiv:math.QA/0209142.
4. Connes A., Moscovici H., Transgression and the Chern character of finite-dimensional $K$-cycles, Comm. Math. Phys. 155 (1993), 103-122.
5. D'Andrea F., Dąbrowski L., Local index formula on the equatorial Podleś sphere, Lett. Math. Phys. 75 (2006), 235-254, arXiv:math.QA/0507337.
6. Engliš M., Falk K., Iochum B., Spectral triples and Toeplitz operators, J. Noncommut. Geom. 9 (2015), 1041-1076, arXiv:1402.3061.
7. Forsyth I., Mesland B., Rennie A., Dense domains, symmetric operators and spectral triples, New York J. Math. 20 (2014), 1001-1020, arXiv:1306.1580.
8. Klimek S., McBride M., d-bar operators on quantum domains, Math. Phys. Anal. Geom. 13 (2010), 357-390, arXiv:1001.2216.
9. Klimek S., McBride M., Rathnayake S., Sakai K., Wang H., Derivations and spectral triples on quantum domains I: Quantum disk, SIGMA 13 (2017), 075, 26 pages, arXiv:1705.04005.
10. Klimek S., McBride M., Rathnayake S., Derivations and spectral triples on quantum domains II: Quantum annulus, Sci. China Math., to appear, arXiv:1710.06257.