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 third-order jets and above is that the connection also be flat and its 'generalised braiding tensor' sigma:Omega(1)circle times(A)Omega(1)->Omega(1)circle times 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(C), the bicrossproduct model quantum spacetime in two dimensions and C-q [SL2] with its 3-dimentional left covariant calculus and q a 4th root of unity.