A class of two-dimensional globally scale-invariant, but not conformally invariant, theories is obtained. These systems are identified in the process of discussing global and local scaling properties of models related by duality transformations, based on non-semisimple isometry groups. The construction of the dual partner of a given model is followed through; non-local as well as local versions of the former are discussed.
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We would like to thank R. Ricci and M. Ro~ek for discussions, and E. Verlinde for a discussion concerning the local representation. We also thank O. Fonarev and Z. Ligeti for help with "Mathematica". S.E., A.G. and E.R. wish to thank the Theory Division at CERN for its warm hospitality. S.E. and E.R. wish to thank the Einstein Center. The work of S.E. is supported in part by the BRF - the Basic Research Foundation. The work of A.G. is supported in part by BSF - American-Israel Bi-National Science Foundation and by an Alon fellowship. The work of E.R. is supported in part by BSF and by the BRF. The work of A.S. is supported in part by BSF.