  The ageing Lie algebra age(d) and especially its local representations for a
dynamical exponent z=2 has played an important r\^ole in the description of
systems undergoing simple ageing, after a quench from a disordered state to the
low-temperature phase. Here, the construction of representations of age(d) for
generic values of z is described for any space dimension d>1, generalising upon
earlier results for d=1. The mechanism for the closure of the Lie algebra is
explained. The Lie algebra generators contain higher-order differential
operators or the Riesz fractional derivative. Co-variant two-time response
functions are derived. Some simple applications to exactly solvable models of
phase separation or interface growth with conserved dynamics are discussed.
