Transposed Poisson structures on Block Lie algebras and superalgebras

Some generalizations of the variety of transposed Poisson algebras

“…Let us formulate the main results of this paper: for commutative Gelfand-Dorfman operad, we determine a set of basis elements up to degree 4. Using these obtained basis elements we show that every commutative Gelfand-Dorfman algebra is F -manifold, and we show that the variety of commutative GD-algebras coincides with the variety of transposed Poisson algebras, which has garnered considerable attention in recent years [1,2,11,12,13]. All mentioned results give…”

Non-associative algebraic structures: classification and structure

“…Other interesting examples of transposed Poisson algebras constructed in a series of papers together with Khrypchenko [143,[147][148][149].…”

Transposed Poisson structures on solvable and perfect Lie algebras

“…In a recent paper by Ferreira, Kaygorodov, Lopatkin a relation between 1 2 -derivations of Lie algebras and transposed Poisson algebras has been established [13]. These ideas were used to describe all transposed Poisson structures on Witt and Virasoro algebras in [13]; on twisted Heisenberg-Virasoro, Schrödinger-Virasoro and extended Schrödinger-Virasoro algebras in [39]; on Schrödinger algebra in (n + 1)-dimensional space-time in [37]; on solvable Lie algebra with filiform nilradical in [1]; on Witt type Lie algebras in [19]; on generalized Witt algebras in [20]; Block Lie algebras in [18,20]; on the Lie algebra of upper triangular matrices in [21]; and on Lie incidence algebras in [22]. It was proved that any complex finite-dimensional solvable Lie algebra admits a non-trivial transposed Poisson structure [23].…”