Transposed Poisson structures on Virasoro-type (super)algebras

Abstract

We explore transposed Poisson structures on the Lie algebra: the deformed twisted Schrödinger-Virasoro algebra $D\left(\lambda \right)$, as well as two Lie superalgebras: the super-BMS₃ algebra and the twisted $N=1$ Schrödinger-Neveu-Schwarz algebra. Initially, we demonstrate the absence of non-trivial transposed Poisson structures on the Lie algebra $D\left(\lambda \right)$ for $\lambda \ne 1$ and provide an example of a transposed Poisson algebra with associative and Lie parts isomorphic to the algebra of triadic extended Laurent polynomials and $D\left(1\right)$. Subsequently, we establish that the super-BMS₃ algebra possesses non-trivial $\frac{1}{2}$-superderivations but lacks a non-trivial transposed Poisson structure. Finally, we prove that the twisted N=1 Schrödinger-Neveu-Schwarz algebra does not have non-trivial $\frac{1}{2}$-superderivations and thus lacks non-trivial transposed Poisson structures.