# Biderivations of the twisted Heisenberg-Virasoro algebra and their applications

## TL;DR

This paper investigates the structure of biderivations in the twisted Heisenberg-Virasoro algebra, revealing non-inner forms and applications to linear maps and algebra structures, advancing understanding of its algebraic properties.

## Contribution

It characterizes non-skew-symmetric biderivations and explores their applications to linear commuting maps and post-Lie algebra structures.

## Key findings

- Identified non-inner, non-skew-symmetric biderivations.
- Characterized linear commuting maps on the algebra.
- Proved all biderivations of the graded algebra are trivial.

## Abstract

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and non-skew-symmetric biderivations. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the twisted Heisenberg-Virasoro algebra are given. It also is proved that every biderivation of the graded twisted Heisenberg-Virasoro left-symmetric algebra is trivial.