Coherent States on Lie Algebras: A Constructive Approach

We generalize the notion of coherent states toarbitrary Lie algebras by making an analogy with the GNSconstruction in C*-algebras. The method is illustratedwith examples of semisimple and nonsemisimple finite-dimensional Lie algebras as well as loopand Kac–Moody algebras. A deformed addition on theparameter space is also introduced simplifying someexpressions and some applications to conformal field theory are pointed out, e.g., differentialoperator and free field realizations found.