Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras and

ABSTRACT

Let $\mathcal{W}$ and $\mathcal{S}$ denote the restricted Cartan-type Lie algebras $W(n,\underline{1})$ and $S(n,\underline{1})$ in a field of characteristic $p>3$, respectively. We prove that each anti-symmetric biderivation of $\mathcal{W}$ and $\mathcal{S}$ is inner. As applications of the anti-symmetric biderivations, it is shown that commuting maps on $\mathcal{W}$ and $\mathcal{S}$ are scalar multiplication maps. Moreover, we prove that the commuting automorphisms and dertivations of $\mathcal{W}$ and $\mathcal{S}$ are the identity mappings and zero mappings, respectively.

KEYWORDS:

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

AMS SUBJECT CLASSIFICATIONS:

• 17B05
• 17B40
• 17B50

Acknowledgements

The authors would like to thank the referee for valuable comments and suggestions on this article.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NNSF of China [grant number 11471090], [grant number 11771069]; NSF of Jilin province [grant number 20170101048JC].