A class of non‐graded left‐symmetric algebraic structures on the Witt algebra

Abstract

We classify the compatible left-symmetric algebraic structures on the Witt algebra satisfying certain non-graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non-graded modules of the Witt algebra and the induced Lie algebras by Novikov-Poisson algebras' approach and Balinskii-Novikov's construction.

Post-Lie Algebra Structures on the Witt Algebra

“…where φ and ̺ are complex-valued functions on Z × Z with ̺ = 0, and ν is a fixed nonzero integer. The motivation of studying this class of non-graded post-Lie algebra structures is inspired by [19], in which a class of non-graded left-symmetric algebraic structures on the Witt algebra has been considered. As in the theory of groups and in the theory of graded Lie algebras, we shall call such non-graded post-Lie algebra to be shifting post-Lie algebra since it with a shifting item.…”

“…We are not going to discuss this problem here. But, inspired by[13,19], we may give two non-trivial examples as follows.…”

Left-symmetric conformal algebras and vertex algebras

“…It should be pointed out that Coeff V a is a compatible Novikov algebra structure on Witt algebra studied in [41].…”

“…Therefore, in the study of vertex algebra, there is a natural question that whether there exist compatible left-symmetric algebra structures on a given formal distribution Lie algebra and how many there are such structures on it. There are some papers on finding the compatible left-symmetric algebra structures on Witt algebra, Virasoro algebra (see [30] and [41]). Their motivations may not be for studying vertex algebras.…”

Compatible left-symmetric algebraic structures on high rank Witt and Virasoro algebras

“…where K is a cental element and θ ∈ C such that θ −1 ∈ Z, Reθ > 0 or Reθ = 0, Imθ > 0. In [16], and further in [13], non-graded CLSAS's on W 1 satisfying that L a L b = f (a, b) L a+b + g (a, b) L a+b+ζ for some ζ ∈ Z \ {0}, and functions f, g on Z × Z, are classified. They turn out to be Novikov algebras [10] Based on these classification results, the CLSAS's on many other Lie algebras containing W 1 or V ir as a subalgebra were considered.…”