Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential Calculus

TL;DR

This paper presents a simplified framework for the maximal prolongation of first-order differential calculi using torsion-free bimodule connections, with applications to quantum homogeneous spaces and quantum Grassmannians.

Contribution

It introduces a concise formula for bimodule maps associated with torsion-free bimodule connections, simplifying the understanding of differential calculus prolongations in quantum spaces.

Findings

Simplified presentation of maximal prolongation using torsion-free bimodule connections.

Explicit formula for bimodule maps in quantum homogeneous spaces.

Application to quantum Grassmannian calculi with uniform relations.

Abstract

We give an unexpectedly simple presentation of the maximal prolongation of a first-order differential calculus in terms of the bimodule map of a torsion-free bimodule connection. We then show that in the quantum homogeneous space case this simplifies even further. More explicitly, we show that the bimodule map associated to a bimodule connection, for any relative left Hopf module endowed with its canonical right module structure, admits a concise formula, given in terms of the adjont action of a Hopf algebra on a bimodule. %{\color{red} We also have the dual tangent space formula.} This is then used to derive sufficient conditions, in terms of the first-order differential forms, for the extendability of a first-order almost-complex structure. These results are applied to the quantum Grassmannian Heckenberger--Kolb calculi, yielding a simple uniform presentation of their degree two anti-holomorphic relations.