Transposed Poisson Structures on the q -Analog Virasoro-like Algebras and q -Quantum Torus Lie Algebras

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