Transposed Poisson structures on Block Lie algebras and superalgebras

Abstract

We describe transposed Poisson algebra structures on Block Lie algebras $\mathcal B(q)$ and Block Lie superalgebras $\mathcal S(q)$, where $q$ is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on $\mathcal B(q)$ are trivial whenever $q\not\in\mathbb Z$, and for each $q\in\mathbb Z$ there is only one (up to an isomorphism) non-trivial transposed Poisson structure on $\mathcal B(q)$. The superalgebra $\mathcal S(q)$ admits only trivial transposed Poisson superalgebra structures for $q\ne 0$ and two non-isomorphic non-trivial transposed Poisson superalgebra structures for $q=0$. As a consequence, new Lie algebras and superalgebras that admit non-trivial ${\rm Hom}$-Lie algebra structures are found.