Transposed Poisson structures on the $q$-analog Virasoro-like algebras and $q$-Quantum Torus Lie algebras

TL;DR

This paper studies transposed Poisson structures on $q$-analog Virasoro-like and $q$-quantum torus Lie algebras, revealing conditions under which non-trivial structures exist depending on whether $q$ is generic or a root of unity.

Contribution

It characterizes the existence of non-trivial transposed Poisson structures on these algebras for different values of $q$, providing explicit structures when possible.

Findings

No non-trivial structures when $q$ is generic for Virasoro-like algebra.

Existence of non-trivial structures when $q$ is a root of unity for Virasoro-like algebra.

Lack of non-trivial transposed Poisson structures on the quantum torus Lie algebra in all cases.

Abstract

We investigate the transposed Poisson structures on both the $q$-analog Virasoro-like algebra and $q$-quantum torus Lie algebra considering the cases where $q$ is generic and where $q$ is a primitive root of unity, respectively. We establish the following results: When $q$ is generic, there are no non-trivial $\frac{1}{2}$-derivations and consequently, no non-trivial transposed Poisson algebra structures exist on the $q$-analog Virasoro-like algebra. Meanwhile, the $q$-quantum torus Lie algebra does possess non-trivial $\frac{1}{2}$-derivations but lacks of a non-trivial transposed Poisson structure. When $q$ is a primitive root of unity, both the $q$-analog Virasoro-like algebra and the $q$-quantum torus Lie algebra possess non-trivial $\frac{1}{2}$-derivations. We present the non-trivial transposed Poisson algebra structure for the $q$-analog Virasoro-like algebra. However, the $q$-quantum torus Lie algebra lacks of a non-trivial transposed Poisson structure.