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

摘要

We find that a compatible graded left-symmetric algebra structure on the Wittalgebra induces an indecomposable module of the Witt algebra with 1-dimensionalweight spaces by its left multiplication operators. From the classification ofsuch modules of the Witt algebra, the compatible graded left-symmetric algebrastructures on the Witt algebra are classified. All of them are simple and theyinclude the examples given by Chapoton and Kupershmidt. Furthermore, weclassify the central extensions of these graded left-symmetric algebras whichgive the compatible graded left-symmetric algebra structures on the Virasoroalgebra. They coincide with the examples given by Kupershmidt.