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

Abstract

Transposed Poisson structures on the Schrödinger algebra in (n + 1)-dimensional spacetime of Schrödinger Lie groups are described. It was proven that the Schrödinger algebra S n in case of n = 2 does not have non-trivial 1 2 -derivations and as it follows it does not admit non-trivial transposed Poisson structures. All 1 2 -derivations and transposed Poisson structures for the algebra S 2 are obtained. Also, we proved that the Schrödinger algebra S 2 admits a non-trivial Hom-Lie structure.

Transposed Poisson Structures

“…On the other hand, there are simple Lie algebras (the Witt algebra W and the Cartan algebra W(1)) that admit non-trivial transposed Poisson structures, 3 From [37], there is a 6-dimensional perfect non-semisimple Lie algebra with non-trivial 1 2 -derivations. 4 From [82], there is a 9-dimensional non-perfect non-solvable Lie algebra with non-trivial but, in these cases, the associative part is not simple (Theorem 19). The last results motivate the following question.…”

Transposed Poisson structures on solvable and perfect Lie algebras

“…In a recent paper by Ferreira, Kaygorodov, Lopatkin a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has been established [13]. 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].…”

Transposed Poisson structures on the Lie algebra of upper triangular matrices

“…In a recent paper by Ferreira, I. Kaygorodov and M. Khrypchenko 136 Kaygorodov, and Lopatkin a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has been established [5]. These ideas were used to describe all transposed Poisson structures on Witt and Virasoro algebras in [5]; on twisted Heisenberg-Virasoro, Schrödinger-Virasoro and extended Schrödinger-Virasoro algebras in [20]; on oscillator Lie algebras in [3]; on Schrödinger algebra in .n C 1/dimensional space-time in [18]; on Witt type Lie algebras in [12]; on generalized Witt algebras in [11] and Block Lie algebras in [10,11]. Any complex finite-dimensional solvable Lie algebra was proved to admit a non-trivial transposed Poisson structure [13].…”