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arXiv · 2202.07331

Levi-Civita connections on quantum spheres

Joakim Arnlind·Kwalombota Ilwale·Giovanni Landi
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Abstract

We introduce $q$-deformed connections on the quantum 2-sphere and 3-sphere, satisfying 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 metric 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.

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BibTeXRIS

Joakim Arnlind, Kwalombota Ilwale, Giovanni Landi. 2022-02-15. Levi-Civita connections on quantum spheres. https://doi.org/10.1007/s11040-022-09431-8

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