Transposed Poisson structures on Block Lie algebras and superalgebras

We describe transposed Poisson algebra structures on Block Lie algebras B ( q ) and Block Lie superalgebras S ( q ) , where q is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on B ( q ) are trivial whenever q ∉ Z , and for each q ∈ Z there is only one (up to an isomorphism) non-trivial transposed Poisson structure on B ( q ) . The superalgebra S ( q ) admits only trivial transposed Poisson superalgebra structures for q ≠ 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 Hom-Lie algebra structures are found.