Maximal Solvable Leibniz Algebras with a Quasi-Filiform Nilradical

Abstract

This article is part of a study on solvable Leibniz algebras with a given nilradical. In this paper, solvable Leibniz algebras, whose nilradical is naturally graded quasi-filiform algebra and the complemented space to the nilradical has maximal dimension, are described up to isomorphism.

Transposed Poisson structures on solvable and perfect Lie algebras

“…Observe that 1 2 -derivations are a particular case of δ-derivations introduced by Filippov (see, for example, [40] and references therein). It is easy to see from definition 3 that [L, L] and Ann(L) are invariant under any 1 2 -derivation of L. Definitions 1 and 3 immediately implies the following key Lemma. 2 -derivation of (L, [−, −]).…”

“…Let (L, [−, −]) be an algebra, φ be a linear map and ϕ be a bilinear map. Then φ is a 1 2 -derivation if it satisfies…”

“…2 -derivation of (L, [−, −]). The basic example of a 1 2 -derivation is the multiplication by a field element. Such 1 2derivations will be called trivial.…”

“…It is proven that any finite-dimensional solvable Lie algebra over an algebraically closed field of zero characteristic admits non-trivial 1 2 -derivations [23]. For the solvable Lie algebra L with non-zero annihilator, there is a non-trivial 1 2 -derivation φ, such that φ(x) = [x, ω 0 ], where ω 0 ∈ Ann [L,L] ([L, L]) and [x, ω 0 ] = λ x ω 0 for all x ∈ L, and λ x ̸ = 0 for some x. Thus, the dimension of the space of 1 2 -derivations of any solvable Lie algebra is at least equal to two.…”

“…Transposed Poisson algebras are related to weak Leibniz algebras [11]. 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].…”