Quantum Jet Bundles

TL;DR

This paper develops a framework for jet bundles over noncommutative algebras with flat connections, incorporating braid relations and Yang-Baxter equations, and explores various algebraic examples.

Contribution

It introduces a novel formulation of jet bundles in noncommutative geometry with flat connections satisfying braid relations, extending classical concepts to quantum and noncommutative settings.

Findings

Formulation of jet bundles over noncommutative algebras with flat connections.

Inclusion of braid relations and Yang-Baxter equations in the structure.

Examples include quantum spacetime and quantum groups.

Abstract

We formulate a notion of jet bundles over a possibly noncommutative algebra $A$ equipped with a torsion free connection. Among the conditions needed for 3rd-order jets and above is that the connection also be flat and its `generalised braiding tensor' $\sigma:\Omega^1\otimes_A\Omega^1\to \Omega^1\otimes_A\Omega^1$ obey the Yang-Baxter equation or braid relations. We also cover the case of jet bundles of a given `vector bundle' over $A$ in the form of a bimodule $E$ with a flat bimodule connection with its braiding $\sigma_E$ obeying the coloured braid relations. Examples include the permutation group $S_3$ with its 2-cycles calculus, $M_2(\Bbb C)$, the bicrossproduct model quantum spacetime in two dimensions and $\Bbb C_q[SL_2]$ for $q$ a 4th root of unity.