Biderivations and linear commuting mappings of $\mathbb{N}$-graded Lie algebras of maximal class

Abstract

Combined with Fialowski's classification for algebras of maximal class and using the first cohomology of these Lie algebras with coefficients in the adjoint module, in this paper we determine the skew-symmetric biderivation spaces for all algebras of maximal class with 1-dimensional homogeneous components m 0 , L 1 and m 2 as the given gradings, up to isomorphism. Different from the usual conclusions, we obtain that a non-simple Lie algebra L 1 only has inner biderivations. As an application, we characterize linear commuting mappings on these algebras.