We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satis- fying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi–Civita connection for a general class of metrics. We also give met- ric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.

Levi–Civita Connections on Quantum Spheres

Landi G.
2022-01-01

Abstract

We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satis- fying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi–Civita connection for a general class of metrics. We also give met- ric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.
2022
Pubblicato
https://link.springer.com/article/10.1007/s11040-022-09431-8
File in questo prodotto:
File Dimensione Formato  
s11040-022-09431-8.pdf

accesso aperto

Tipologia: Documento in Versione Editoriale
Licenza: Creative commons
Dimensione 413.27 kB
Formato Adobe PDF
413.27 kB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11368/3038347
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact