Transposed Poisson structures on Galilean and solvable Lie algebras

Transposed Poisson structures on solvable and perfect Lie algebras

“…It is proven that any finite-dimensional solvable Lie algebra over an algebraically closed field of zero characteristic admits non-trivial 1 2 -derivations [23]. For the solvable Lie algebra L with non-zero annihilator, there is a non-trivial 1 2 -derivation φ, such that φ(x) = [x, ω 0 ], where ω 0 ∈ Ann [L,L] ([L, L]) and [x, ω 0 ] = λ x ω 0 for all x ∈ L, and λ x ̸ = 0 for some x.…”

“…These ideas were used to describe all transposed Poisson structures on Witt and Virasoro algebras in [13]; on twisted Heisenberg-Virasoro, Schrödinger-Virasoro and extended Schrödinger-Virasoro algebras in [39]; on Schrödinger algebra in (n + 1)-dimensional space-time in [37]; on solvable Lie algebra with filiform nilradical in [1]; on Witt type Lie algebras in [19]; on generalized Witt algebras in [20]; Block Lie algebras in [18,20]; on the Lie algebra of upper triangular matrices in [21]; and on Lie incidence algebras in [22]. 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].…”

Transposed Poisson Structures

“…There are no non-trivial transposed Poisson structures defined on a complex finite-dimensional semisimple Lie algebra [20,Corollary 9]. On the other hand, the Schrödinger algebra, which is the semidirect product of a simple and a nilpotent algebra, also does not have non-trivial transposed Poisson structures (other perfect non-simple Lie algebras without non-trivial 1 2 -derivations, also known as Galilean algebras, can be found in [38]). Theorem 9 states that each finite-dimensional nilpotent Lie algebra has a non-trivial transposed Poisson structure ( 12 -derivations, 1 2 -biderivations).…”

“…Theorem 9 states that each finite-dimensional nilpotent Lie algebra has a non-trivial transposed Poisson structure ( 12 -derivations, 1 2 -biderivations). Results from [38] state that each finite-dimensional solvable non-nilpotent Lie algebra has a non-trivial transposed Poisson structure ( 12 -derivations, 1 2 -biderivations). The last results motivate the following question.…”

Transposed Poisson structures on Schrodinger algebra in (n+1)-dimensional space-time

“…More recently, a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has not only been established [11], but also between 1 2 -biderivations and transposed Poisson algebras [32]. These ideas were used for describing all transposed Poisson structures on the Witt algebra which was one of the first examples of non-trivial transposed Poisson algebras [11], the Virasoro algebra [11], the algebra W (a, b) [11], the thin Lie algebra [11], the twisted Heisenberg-Virasoro algebra [32], the Schrödinger-Virasoro algebra [32], the extended Schrödinger-Virasoro algebra [32], the 3-dimensional Heisenberg Lie algebra [32], Block Lie algebras and superalgebras [17], Witt type algebras [19], oscillator Lie algebras [4] and Galilean and solvable Lie algebras [18]. A list of actual open questions on transposed Poisson algebras was given in [4].…”