  Let 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.
