2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/128631Let g denote a Lie algebra over a field of characteristic zero, and let T(g) denote the tensor product of g with a ring of truncated polynomials. The Lie algebra T(g) is called a truncated current Lie algebra, or in the special case when g is finite-dimensional and semisimple, a generalized Takiff algebra. In this paper a highest-weight theory for T(g) is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of T(g) for a wide class of Lie algebras g, including the symmetrizable Kac-Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov form.42 pages. An extract from the author's PhD thesis. See also: http://www.maths.usyd.edu.au/u/benw/Representation TheoryMathematical Physics17B10; 17B65; 17B67; 17B68text