Coherent states on Lie algebras: A constructive approach

We generalize the notion of coherent states to arbitrary Lie algebras by ma king an analogy with the GNS construction in C*-algebras. The method is ill ustrated with examples of semisimple and nonsemisimple finite-dimensional L ie algebras as well as loop and Kac-Moody algebras. A deformed addition on the parameter space is also introduced simplifying some expressions and som e applications to conformal field theory are pointed out, e.g., differentia l operator and free field realizations found.