The algebraic and geometric classification of transposed Poisson algebras

Abstract

The algebraic and geometric classification of all complex 3-dimensional transposed Poisson algebras is obtained. Also we discuss special 3-dimensional transposed Poisson algebras.

Transposed Poisson Structures

“…The variety of transposed Poisson algebras gives a subvariety in C 2n 3 . The geometric classification of complex 3-dimensional transposed Poisson algebras was given in [6]. Neretin, in his paper [65], considered n-dimensional associative, commutative, and Lie algebras, and gave the upper bound for the dimensions of their geometric varieties.…”

Transposed Poisson structures on Galilean and solvable Lie algebras

“…Moreover, the algebra L 2 is a perfect algebra (i.e., [L 2 , L 2 ] = L 2 ) and admits non-trivial 1 2 -derivations. The existence of algebras with this property gives a particular answer to [4,Questions 2 and 5].…”

“…Any complex finite-dimensional solvable Lie algebra was proved to admit a non-trivial transposed Poisson structure [21]. The algebraic and geometric classification of 3-dimensional transposed Poisson algebras was given in [4]. For the list of actual open questions on transposed Poisson algebras, see [5].…”

“…we have that α4 2 = α 4 3 = β 4 k = 0, 1 ≤ k ≤ m. Considering the following relation2mϕ(x 0 ) = 2ϕ([h, x 0 ]) = [ϕ(h), x 0 ] + [h, ϕ(x 0 )] = [α 1 1 h, x 0 ] + [h, α 4 1 e + β 4 0 x 0 ] = 2α 4 1 e + m(α 1 1 + β 4 0 )x 1 , Thus, ϕ(x 0 ) = α 1 1 x 0 . Then from ϕ(x k ) = ϕ([x k−1 , f ]), we obtain that ϕ(x k ) = α 1 1 x k for 1 ≤ k ≤ m.Hence, any 1 2 -derivation of L m is trivial.…”

Transposed Poisson structures on solvable and perfect Lie algebras

“…It was proved that any complex finite-dimensional solvable Lie algebra admits a non-trivial transposed Poisson structure [23]. The algebraic and geometric classification of 3dimensional transposed Poisson algebras was given in [5]. For the list of current open questions on transposed Poisson algebras, see [6].…”