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

TL;DR

This paper classifies transposed Poisson structures on Schr"odinger algebras in (n+1)-dimensional space-time, showing non-trivial structures only exist for the 2-dimensional case and establishing a non-trivial Hom-Lie structure there.

Contribution

It provides a complete description of transposed Poisson structures on Schr"odinger algebras, highlighting the special case of the 2-dimensional algebra and its non-trivial Hom-Lie structure.

Findings

Schr"odinger algebra $\\mathcal{S}_n$ for $n eq 2$ has no non-trivial $\frac{1}{2}$-derivations.

All $\frac{1}{2}$-derivations and transposed Poisson structures are classified for $\mathcal{S}_2$.

The algebra $\mathcal{S}_2$ admits a non-trivial Hom-Lie structure.

Abstract

Transposed Poisson structures on the Schr\"{o}dinger algebra in $(n+1)$-dimensional space-time of Schr\"{o}dinger Lie groups are described. It was proven that the Schr\"{o}dinger algebra $\mathcal{S}_{n}$ in case of $n\neq 2$ does not have non-trivial $\frac{1}{2}$-derivations and as it follows it does not admit non-trivial transposed Poisson structures. All $\frac{1}{2}$-derivations and transposed Poisson structures for the algebra $\mathcal{S}_{2}$ are obtained. Also, we proved that the Schr\"{o}dinger algebra $\mathcal{S}_{2}$ admits a non-trivial ${\rm Hom}$-Lie structure.