Central Extension of Scaling Poincar\'e Algebra

We discuss the possibility of a central extension of the Poincar\'e algebra and the scaling Poincar\'e algebra. In more than two space-time dimensions, all the central extensions are trivial and can be removed. In two space-time dimensions, both the Poincar\'e algebra and the scaling Poincar\'e algebra have distinct non-trivial central extensions that cannot be removed. In higher dimensions, the central charges between dilatation and global $U(1)$ symmetry may not be removed. Based on these central extensions, we give some examples of projective representations of the (scaling) Poincar\'e symmetry in two dimensions.