Transposed Poisson structure on the Witt-type algebra W (a, -1): Derivations, Automorphisms, and Rota--Baxter operators

In this paper, we provide a comprehensive study of the structural properties of the transposed Poisson algebra W (a, -1). We classify several types of linear maps, including derivations, local derivations, quasi-derivations, and -derivations, showing that non-trivial -derivations exist only for =1 and =12. Furthermore, we describe the groups of automorphisms, local automorphisms, 2-local automorphisms, and quasi-automorphisms. We also investigate Rota--Baxter operators of weight 1 on W (a, -1). Specifically, we classify operators that are homogeneous with respect to both the standard Z-grading and a Z₂-grading, establishing a rigidity result for the latter case. Finally, we classify all W-compatible Novikov--Poisson structures, demonstrating that the associative product on the Witt algebra is universally compatible with its known Novikov structures.