Classification of Graded Left-symmetric Algebra Structures on Witt and Virasoro Algebras

TL;DR

This paper classifies graded left-symmetric algebra structures on the Witt algebra and their extensions to the Virasoro algebra, revealing all such structures are simple and include known examples.

Contribution

It provides a complete classification of compatible graded left-symmetric algebra structures on Witt and Virasoro algebras, including their central extensions.

Findings

All compatible structures are simple and include Chapoton and Kupershmidt examples.

Classified central extensions correspond to known Virasoro algebra structures.

The classification links module theory with algebra structures on Witt and Virasoro algebras.

Abstract

We find that a compatible graded left-symmetric algebra structure on the Witt algebra induces an indecomposable module of the Witt algebra with 1-dimensional weight spaces by its left multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebra structures on the Witt algebra are classified. All of them are simple and they include the examples given by Chapoton and Kupershmidt. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebra structures on the Virasoro algebra. They coincide with the examples given by Kupershmidt.