On Local Automorphisms of $\mathfrak{sl}_2$

TL;DR

This paper characterizes local automorphisms of the Lie algebra sl_2, showing they coincide with automorphisms and anti-automorphisms, and extends the result to anti-automorphisms of sl_n for ngeq 3.

Contribution

It precisely identifies local automorphisms of sl_2 as automorphisms and anti-automorphisms, and demonstrates anti-automorphisms are local automorphisms for sl_n when ngeq 3.

Findings

LAut(sl_2) equals Aut^{b1}(sl_2)

Anti-automorphisms are local automorphisms of sl_n for ngeq 3

Provides a complete description of local automorphisms for these Lie algebras

Abstract

We establish that the set of local automorphisms $\textrm{LAut}(\mathfrak{sl}_2)$ is the group $\textrm{Aut}^{\pm}(\mathfrak{sl}_2)$ of all automorphisms and anti-automorphisms. For $n\geq 3$ we prove that anti-automorphisms are local automorphisms of $\mathfrak{sl}_n$.