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

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... rators 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.