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

Abstract

In this paper, we provide a comprehensive study of the structural properties of the transposed Poisson algebra $\mathcal{W}(a,-1)$. We classify several types of linear maps, including derivations, local derivations, quasi-derivations, and $\hat{I}´$-derivations, showing that non-trivial $\hat{I}´$-derivations exist only for $\hat{I}´=1$ and $\hat{I}´=\frac{1}{2}$. 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 $\mathcal{W}(a,-1)$. Specifically, we classify operators that are homogeneous with respect to both the standard $\mathbf{Z}$-grading and a ${\mathbf{Z}}_2$-grading, establishing a rigidity result for the latter case. Finally, we classify all $\mathbf{W}$-compatible Novikov--Poisson structures, demonstrating that the associative product on the Witt algebra is universally compatible with its known Novikov structures.