PreprintHighestWeight Theory for Truncated Current Lie AlgebrasBen WilsonAbstractLet 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 finitedimensional and semisimple, a generalized Takiff algebra. In this paper a highestweight 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 KacMoody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov form.Keywords: current algebra; takiff algebra; verma module; triangular decomposition; exppolynomial module.
