Speciality of transposed Poisson algebras

Prove that every transposed Poisson algebra is special, namely establish that any transposed Poisson algebra admits an embedding into an appropriate differential Poisson algebra.

Background

The authors highlight a central conjectural property of transposed Poisson algebras—speciality—framed as an open problem drawn from the literature on GD-algebras and their generalizations. Proving speciality would position transposed Poisson algebras within the well-understood framework of differential Poisson algebras.

Establishing this embedding is expected to facilitate the construction of bases for free transposed Poisson algebras by leveraging known results for special GD-algebras.

References

Many open problems associated with GD-algebras and their generalizations are stated in [5]. One of these problems is to prove that any transposed Poisson algebra is special, i.e., it can be embedded into appropriate differential Poisson algebra.

— On the free metabelian transposed Poisson and F-manifold algebras (2408.05512 - Abdukhalikov et al., 10 Aug 2024) in Section 1. Introduction