Representations and O-operators of Hom-(pre)-Jacobi-Jordan algebras

Mock-Lie bialgebras and mock-Lie analogue of the classical Yang-Baxter equation

“…In this section, we provide some preliminaries about mock-Lie algebras and left mock-pre-Lie algebras. Our main references are [2,3,11,18]. Definition 2.1.…”

Lie 3-algebras and deformations of relative Rota-Baxter operators on 3-Lie algebras

“…Definition 2.10. [2] A representation of a Hom-Jacobi-Jordan algebra (A, * , α) is a triple (V, ρ, φ) where V is a vector space, φ ∈ gl(V ) and ρ : A → gl(V ) is a linear map such that the following equalities hold for all x, y ∈ A,…”

“…Example 2.12. [2] Let (A, * , α) be a Hom-Jacobi-Jordan algebra and (B, α) be a Hom-ideal of (A, * , α). Set ρ(a)b := a * b for all (a, b) ∈ A × B, then (B, α) is a representation of (A, * , α).…”

“…Definition 5.1. [2] Let (V, ρ, φ) be a representation of a Hom-Jacobi-Jordan algebra (A, * , α). A linear map T : V → A is called a relative Rota-Baxter operator with respect to (V, ρ, φ) if it satisfies…”

“…Due to the importance of Hom-algebras in several domains, the twisted generalization of Jacobi-Jordan algebra called Hom-Jacobi-Jordan algebra is initiated in [8] while its representation theory is introduced in [2]. Roughly, a Hom-type of a given algebra is obtained by a certain twisting of the defining identities by a twisting map, in such a way that when this twisting map is the identity map, then one recovers the original algebra.…”