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 SLq(2).
Lingua originaleInglese
pagine (da-a)479-544
Numero di pagine66
RivistaJournal of Algebra
Volume661
DOI
Stato di pubblicazionePubblicato - 2025

Keywords

  • Bicovariant bimodules
  • Non-equivariant connections
  • Non-equivariant non-central metrics
  • Noncommutative Riemannian geometry
  • Rational morphisms
  • Tensor products of noncommutative connections

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