-(bi)derivations and transposed Poisson algebra structures on Lie algebras. Issue 22 (21st December 2022)

• Record Type: Journal Article
• Authors: Yuan, Lamei; Hua, Qianyi
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 22(2022)
• Page Start: 7672
• Page End: 7701
• Publication Date: 2022-12-21
• Subjects: Transposed Poisson algebras -- δ-biderivations -- twisted Heisenberg–Virasoro algebra -- Schrödinger–Virasoro algebra -- current Lie algebras -- Heisenberg Lie algebra; 17A30 -- 17A60 -- 17B68 -- 17B70; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2021.2003287
• Languages: English
• ISSNs: 0308-1087

Abstract

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 1 2 -biderivations and transposed Poisson algebras. Second, we compute 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 1 2 -derivations. Hence they have neither nonzero 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.