-(bi)derivations and transposed Poisson algebra structures on Lie algebras

In the present paper, we introduce the notion of a δ-biderivation. First, we provide some properties of δ-biderivations and illustrate their applications. In particular, we establish a close relationship between 12-biderivations and transposed Poisson algebras. Second, we compute 12-derivations on the twisted Heisenberg–Virasoro, Schrödinger–Virasoro, extended Schrödinger–Virasoro and twisted Schrödinger–Virasoro algebras, respectively. It turns out that they have no nontrivial 12-derivations. Hence they have neither nonzero 12-biderivations nor nontrivial transposed Poisson algebra structures. Third, we classify transposed Poisson algebra structures on the Heisenberg and some current Lie algebras. This enables us to provide examples of Lie algebras having nontrivial transposed Poisson algebra structures.