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

Abstract

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 $\frac{1}{2}$-biderivations and transposed Poisson algebras. Second, we compute $\frac{1}{2}$-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 $\frac{1}{2}$-derivations. Hence they have neither nonzero $\frac{1}{2}$-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.

Keywords:

• Transposed Poisson algebras
• δ-biderivations
• twisted Heisenberg–Virasoro algebra
• Schrödinger–Virasoro algebra
• current Lie algebras
• Heisenberg Lie algebra

2020 Mathematics Subject Classifications:

• 17A30
• 17A60
• 17B68
• 17B70

Acknowledgments

The authors thank the referee for many useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by the National Natural Science Foundation grants of China [grant number 11301109].