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Metric compatibility and Levi-Civita Connections on Quantum Groups
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abstract
Arbitrary connections on a generic Hopf algebra $H$ are studied and shown to extend to connections on tensor fields. On this ground a general definition of metric compatible connection is proposed. This leads to a sufficient criterion for the existence and uniqueness of the Levi-Civita connection, that of invertibility of an $H$-valued matrix. Provided invertibility for one metric, existence and uniqueness of the Levi-Civita connection for all metrics conformal to the initial one is proven. This class consists of metrics which are neither central (bimodule maps) nor equivariant, in general. For central and bicoinvariant metrics the invertibility condition is further simplified to a metric independent one. Examples include metrics on $SL_q(2)$.
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Cited by 1 Pith paper
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On the existence of noncommutative Levi-Civita connections in derivation based calculi
Weak symmetry of the hermitian form, dρ=0, is a necessary condition for existence of Levi-Civita connections in derivation based calculi, and sufficient for free modules with a dual basis.
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