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

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