Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras H(n; 1)

Abstract

Let $\mathbb{F}$ be a field of characteristic p > 3 and $\mathcal{H}$ be the restricted Cartan-type Lie algebras $H(n,\underset{¯}{1})$. We prove that each skew-symmetric biderivation of $\mathcal{H}$ is inner. As applications of the skew-symmetric biderivations, it is shown that commuting mappings on $\mathcal{H}$ are scalar multiplication mappings. Moreover, we prove that the commuting automorphisms and dertivations of $\mathcal{H}$ are the identity mappings and zero mappings, respectively.

Keywords:

• Restricted Cartan-type Lie algebras
• biderivations
• linear commuting maps

2010 Mathematics Subject Classification:

• 17B05
• 17B40
• 17B50

Additional information

Funding

Supported by NNSF of China (Nos. 11471090 and 11771069), NSF of Jilin province (No. 20170101048JC) and the project of jilin province department of education (No. JJKH20180005K).