Derivations and automorphism groups of completed Witt Lie algebra

TL;DR

This paper introduces the completed Witt Lie algebra, fully describes its derivation algebra and automorphism group, and explores its cohomology and conjugacy classes, revealing its structural properties.

Contribution

It provides a complete description of the derivation algebra, automorphism group, and conjugacy classes of the completed Witt Lie algebra, a new simple Lie algebra.

Findings

First cohomology group with coefficients in the adjoint module is trivial

Automorphism group is fully characterized

Conjugacy classes under automorphisms are determined

Abstract

In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a byproduct, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad-locally finite element.