Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 070, 21 pages      arXiv:2202.09842

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

Alessandro Carotenuto and Réamonn Ó Buachalla
Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech Republic

Received April 14, 2022, in final form September 18, 2022; Published online October 01, 2022

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 $\mathcal{O}_q(G/L_S)$ 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 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 $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.

Key words: quantum groups; noncommutative geometry; bimodule connections; quantum principal bundles; quantum flag manifolds; complex geometry.

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