Levi-Civita connection on the irreducible quantum flag manifolds

TL;DR

This paper classifies covariant metrics on quantum homogeneous spaces and proves the existence and uniqueness of Levi-Civita connections for these metrics on irreducible quantum flag manifolds, extending previous results.

Contribution

It provides a classification of covariant metrics and establishes the existence and uniqueness of Levi-Civita connections on quantum flag manifolds, generalizing prior work on quantum projective spaces.

Findings

Unique quantum symmetric covariant metric exists on Heckenberger--Kolb calculi.

Levi-Civita connection exists and is unique for any real covariant metric.

Results extend classical Levi-Civita connection to quantum flag manifolds.

Abstract

We classify covariant metrics (in the sense of Beggs and Majid) on a class of quantum homogeneous spaces. In particular, our classification implies the existence of a unique (up to scalar) quantum symmetric covariant metric on the Heckenberger--Kolb calculi for the quantized irreducible flag manifolds. Moreover, we prove the existence and uniqueness of Levi-Civita connection for any real covariant metric for the Heckenberger--Kolb calculi. This generalizes Matassa's result for the quantum projective spaces.