Central extensions, derivations, and automorphisms of semi-direct sums of the Witt algebra with its intermediate series modules

TL;DR

This paper thoroughly investigates the structure of Lie algebras formed by semi-direct sums of the Witt algebra and its intermediate series modules, focusing on their central extensions, derivations, and automorphisms, using grading techniques.

Contribution

It completes the classification of central extensions, derivations, and automorphisms for these semi-direct sum Lie algebras, employing methods based on the internal grading of the Witt algebra.

Findings

Classification of central extensions completed

Derivations and automorphisms characterized

Grading techniques applicable to broader classes

Abstract

Lie algebras formed via semi-direct sums of the Witt algebra $\text{Der}(\mathbb{C}[t,t^{-1}])$ and its modules have become increasingly prominent in both physics and mathematics in recent years. In this paper, we complete the study of (Leibniz) central extensions, derivations and automorphisms of the Lie algebras formed from the semi-direct sum of the Witt algebra and its indecomposable intermediate series modules (that is, graded modules with one-dimensional graded components). Our techniques exploit the internal grading of the Witt algebra, which can be applied to a wider class of graded Lie algebras.