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

DOI10.3842/SIGMA.2022.070arXiv2202.09842MaRDI QIDQ6391643

Alessandro Carotenuto, Réamonn Ó Buachalla

Publication date: 20 February 2022

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 $Omeg{a}_q^1(G/L_S)$. 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.

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Quantum groups and related algebraic methods applied to problems in quantum theory (81R50) Noncommutative differential geometry (46L87) Noncommutative geometry in quantum theory (81R60) Quantizations, deformations for selfadjoint operator algebras (46L65) Hopf algebras and their applications (16T05)

This page was built for publication: Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6391643)