Summary: The ageing Lie algebra \(\mathfrak {age}(d)\) and especially its local representations for a dynamical exponent \(z = 2\) have played an important role 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 \(\mathfrak {age}(d)\) for generic values of \(z\) is described for any space dimension \(d > 1\), generalizing 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. Covariant two-time response functions are derived. Some simple applications to the exactly solvable models of phase separation or interface growth with conserved dynamics are discussed.