The algebraic and geometric classifications of complex 3-dimensional transposed Formula: see text-Poisson algebras are given. Namely, we prove that the variety of complex 3-dimensional transposed Formula: see text-Poisson algebras has dimension 9 and 7 irreducible components for Formula: see text the variety of complex 3-dimensional transposed (-1)-Poisson algebras has dimension 9 and 5 irreducible components; the variety of complex 3-dimensional transposed 0-Poisson algebras has dimension 11 and 4 irreducible components; the variety of complex 3-dimensional transposed Formula: see text-Poisson algebras has dimension 9 and 5 irreducible components; and the variety of complex 3-dimensional transposed 1-Poisson algebras has dimension 10 and 3 irreducible components. In particular, we find the first example (in an ”adequate” variety) of deformations (and degenerations) between two simple transposed anti-Poisson algebras. As a byproduct, we obtain the algebraic and geometric classification of complex 3-dimensional transposed scalar-Poisson algebras, i.e., algebras that are in the intersection of all varieties of transposed Formula: see text-Poisson algebras.
Abdelwahab et al. (2026) studied this question.