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

Xiaoli Kong, Hongjia Chen, Chengming Bai

Arxiv ID: 0706.4229•Last updated: 11/18/2020

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.

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