Arkiv för Matematik

Volume 59 (2021)

Number 1

Local and $2$-local automorphisms of simple generalized Witt algebras

Pages: 1 – 10

DOI: https://dx.doi.org/10.4310/ARKIV.2021.v59.n1.a1

Authors

Yang Chen (Mathematics Postdoctoral Research Center, Hebei Normal University, Shijiazhuang, Heibei, China)

Kaiming Zhao (Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada; and School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei, China)

Yueqiang Zhao (School of Mathematics and Statistics, Xinyang Normal University, Xinyang, Henan, China)

Abstract

In this paper, we prove that every invertible $2$-local or local automorphism of a simple generalized Witt algebra over any field of characteristic $0$ is an automorphism. Furthermore, every $2$-local or local automorphism of Witt algebras $W_n$ is an automorphism for all $n \in \mathbb{N}$. But some simple generalized Witt algebras indeed have $2$-local (and local) automorphisms that are not automorphisms.

Keywords

Lie algebra, generalized Witt algebra, automorphism, local automorphism, $2$-local automorphism

2010 Mathematics Subject Classification

17B05, 17B40, 17B66

This research is partially supported by NSFC (11871190) and NSERC (311907-2015).

Received 1 April 2020

Received revised 24 October 2020

Accepted 6 November 2020