Metric compatibility and Levi-Civita Connections on Quantum Groups

TL;DR

This paper develops a general framework for metric compatibility and Levi-Civita connections on quantum groups, providing criteria for their existence and uniqueness, with applications to metrics on $SL_q(2)$.

Contribution

It introduces a new definition of metric compatible connections on Hopf algebras and establishes conditions for the existence and uniqueness of Levi-Civita connections.

Findings

A criterion based on invertibility of an $H$-valued matrix for Levi-Civita connections.

Existence and uniqueness for metrics conformal to a given metric.

Simplified invertibility conditions for central and bicoinvariant metrics.

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 $SL_q(2)$.