Transposed Poisson structures on quasi-filiform Lie algebras of maximum length

TL;DR

This paper investigates the structure of transposed Poisson algebras on quasi-filiform Lie algebras of maximum length, using $rac{1}{2}$-derivations to construct and analyze these algebraic structures.

Contribution

It introduces a method to construct non-trivial transposed Poisson algebras on specific Lie algebras using $rac{1}{2}$-derivations, expanding understanding of their algebraic properties.

Findings

Constructed new transposed Poisson algebras on quasi-filiform Lie algebras

Established a commutative associative multiplication compatible with the Lie algebra

Provided explicit examples of non-trivial transposed Poisson structures

Abstract

This article will discussing on $\frac{1}{2}$-derivations of quasi-filiform Lie algebras of maximum length. The non-trivial transposed Poisson algebras with the quasi-filiform Lie algebras of maximum length are constructed by using $\frac{1}{2}$-derivations of Lie algebras. We have established commutative associative multiplication to construct a transposed Poisson algebra with an associated given Lie algebra.