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

  • Roman Lubkov
  • Ilya Shiringovskiy
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Abstract

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 δ-derivations, showing that non-trivial δ-derivations exist only for δ=1 and δ=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 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.

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Cite

BibTeXRIS
BibTeX
@misc{lubkov2026transposed,
  title = {{Transposed Poisson structure on the Witt-type algebra $\mathcalW(a,-1)$: Derivations, Automorphisms, and Rota--Baxter operators}},
  author = {Roman Lubkov and Ilya Shiringovskiy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2606.20584},
  eprint = {2606.20584},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.20584},
}
RIS
TY  - UNPB
TI  - Transposed Poisson structure on the Witt-type algebra 𝒲(a,−1): Derivations, Automorphisms, and Rota--Baxter operators
AU  - Roman Lubkov
AU  - Ilya Shiringovskiy
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2606.20584
UR  - https://arxiv.org/abs/2606.20584
AB  - 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 δ-derivations, showing that non-trivial δ-derivations exist only for δ=1 and δ=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 𝒲(a,−1). Specifically, we classify operators that are homogeneous with respect to both the standard Z-grading and a Z_2-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.
ER  -