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arXiv · 2310.00624

Transposed Poisson structures on solvable and perfect Lie algebras

Ivan Kaygorodov·Abror Khudoyberdiyev
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Abstract

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e., on one-dimensional solvable extensions of the $(2n+1)$-dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on $(n+1)$-dimensional solvable extensions of the $(2n+1)$-dimensional Heisenberg algebra; and on $n$-dimensional solvable extensions of the $n$-dimensional algebra with the trivial multiplication. We also gave an answer to one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira, and Kaygorodov. Namely, we found a finite-dimensional Lie algebra with non-trivial $\frac{1}{2}$-derivations, but without non-trivial transposed Poisson algebra structures.

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Ivan Kaygorodov, Abror Khudoyberdiyev. 2023-10-01. Transposed Poisson structures on solvable and perfect Lie algebras. https://doi.org/10.1088/1751-8121%2Fad1620

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