Highest-weight theory for truncated current Lie algebras

In this paper a highest-weight theory for the truncated current Lie algebra gˆgˆ=g⊗kk[t]/tN+1k[t] is developed when the underlying Lie algebra gg possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of gˆ for a wide class of Lie algebras gg, including the symmetrizable Kac–Moody Lie algebras, the Heisenberg algebra, and the Virasoro algebra. This is achieved through a study of the Shapovalov form.