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

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.