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

Abstract

It was recently shown (by the second author and Díaz García, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold 𝒪q(𝐺/𝐿ₛ) admits a unique 𝒪q(𝐺)-covariant connection with respect to the Heckenberger-Kolb differential calculus Ω¹q(𝐺/𝐿ₛ). 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 Díaz García. 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 𝒪q(𝐺), and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.