Quantum Jet Bundles

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, $M_2(\mathbb{C})$, the bicrossproduct model quantum spacetime in two dimensions and ${\mathbb{C}}_q[S{L}_2]$ for $q$ a 4th root of unity.