Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

TL;DR

This paper demonstrates that all relative line modules over irreducible quantum flag manifolds possess bimodule connections with invertible bimodule maps, extending previous covariant connection results using quantum principal bundle frameworks.

Contribution

It establishes the bimodule nature and invertibility of the associated bimodule maps for these connections, applying general quantum principal bundle theory to specific quantum flag manifold structures.

Findings

Connections are bimodule with invertible maps

Explicit bimodule maps are expressed via quantum determinants

Uses categorical equivalence for relative Hopf modules

Abstract

It was recently shown (by the second author and D\'{i}az Garc\'{i}a, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $\Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and D\'{i}az Garc\'{i}a. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.