Biderivations and commuting linear maps on Hom-Lie algebras

Abstract

The purpose of this paper is to determine skew-symmetric biderivations Bider s (L,V ) and commuting linear maps Com(L,V ) on a Hom-Lie algebra (L, α) having their ranges in an (L, α)-module (V, ρ, β ), which are both closely related to Cent(L,V ), the centroid of (V, ρ, β ). Specifically, under appropriate assumptions, every δ ∈ Bider s (L,V ) is of the form δ (x, y) = β −1 γ([x, y]) for some γ ∈ Cent(L,V ), and Com(L,V ) coincides with Cent(L,V ). Besides, we give the algorithm for describing Bider s (L,V ) and Com(L,V ) respectively, and provide several examples.

Biderivations of Hom-Lie algebras and superalgebras

“…Since both the q-deformed Witt and W (2, 2) algebras are not multiplicative, we can not use the definition and related results given in [22] to explore biderivations of these two algebras. Hence, we need the following more general concepts.…”

“…Recently, the notion of a skew-symmetric biderivation from a multiplicative Hom-Lie algebra L to its Hommodule M was introduced and studied in [22], extending some concepts and results developed in [9]. However, the definition is only applicable to multiplicative Hom-Lie algebras, and does not cover symmetric or more general biderivations.…”

Matching BiHom-Rota-Baxter Algebras and Related Structures

“…The Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov [1]. Further research on Hom-Lie algebras could be found in [2][3][4][5][6][7] and references cited therein. As a generalization of Hom-Lie algebra, Graziani etc.…”