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Indecomposable but reducible representations of several Lie algebras of localscale-transformations, including the Schr\"odinger and conformal Galileanalgebras, and some of their applications in physical ageing are reviewed. Thephysical requirement of the decay of co-variant two-point functions for largedistances is related to analyticity properties in the coordinates dual to thephysical masses or rapidities.

Physical ageing and new representations of some Lie algebras of local scale-invariance