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.