Quantum jet Hopf algebroids by cotwist

TL;DR

This paper introduces a cotwist construction for Hopf algebroids, enabling the quantisation of the base and relating jet bundles to noncommutative geometry, expanding the algebraic framework for quantum structures.

Contribution

It presents a dual cotwist construction for Hopf algebroids, connecting jet bundles with noncommutative bases and extending previous twisting methods to quantum settings.

Findings

Jet bundle algebra $J(B)$ forms a Hopf algebroid.

The cotwist construction allows deformation to noncommutative jet Hopf algebroids.

Identification of $J^k(B)$ with $J^1(B_k)$ in the noncommutative calculus setting.

Abstract

We introduce a cotwist construction for Hopf algebroids that also entails cotwisting or `quantisation' of the base and which is dual to a previous twisting construction of P. Xu. Whereas the latter applied the construction to the algebra of differential operators on a classical base $B$, we show that the dual of this is the algebra of sections $J(B)$ of the jet bundle and hence that the latter forms a Hopf algebroid. This is constructed for commutative algebras $B$ in a pro-object setting via quotients of the pair Hopf algebroid $B\otimes B$ and can then be deformed by our cotwist construction to give a possibly noncommutative jet Hopf algebroid over a noncommutative base. We also observe in the commutative case that $J^k(B)$ for jets of order $k$ can be identified with $J^1(B_k)$ where $B_k$ denotes $B$ equipped with a certain noncommutative first order differential calculus.