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 nontrivial [Formula: see text]-derivations and consequently, no nontrivial 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 nontrivial [Formula: see text]-derivations but lacks of a nontrivial 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 nontrivial [Formula: see text]-derivations. We present the nontrivial 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 nontrivial transposed Poisson structure.