The N-derivation is a natural generalization of derivation and triple derivation. Let L be a finitely generated Lie algebra graded by a finite-dimensional Cartan subalgebra. In this paper, a sufficient condition for the Lie N-derivation algebra of Lcoinciding with the Lie derivation algebra of L is given. As applications, any N-derivation of the SchrSdinger- Virasoro algebra, generalized Witt algebra, Kac-Moody algebra or their Borel subalgebra is a derivation.
Let L be the derivation Lie algebra of C[t1^±1 , t2^±1 ]. Given a triangle decomposition L = L＋ η ＋ L-, we define a nonsingular Lie algebra homomorphism φ ： L＋ → C and the universal Whittaker L-module We of type φ. We obtain all Whittaker vectors and submodules of We. Moreover, all simple Whittaker L-modules of type φ are determined.
