Highest weight modules over pre-exp-polynomial Lie algebras

In this paper, we introduce pre-exp-polynomial Lie algebras which include loop algebras, Virasoro-like algebras and some quantum torus Lie algebras. We study “highest weight” representations of these Zn+1-graded Lie algebras. More precisely, we show that non-graded and graded irreducible highest weight modules with the same highest weight simultaneously have all finite-dimensional weight spaces or not, and they have all finite-dimensional weight spaces if and only if the highest weight is an exp-polynomial “highest weight”. We also show that non-graded and graded highest weight Verma modules with the same highest weight are simultaneously irreducible or not, and we determine necessary and sufficient conditions for a Verma module to be irreducible.