On a pre-Jacobi-Jordan algebra: relevant properties and double construction

Abstract

We introduce a pre-Jacobi-Jordan algebras and study some relevant properties such as bimodules, matched pairs. Besides, we established a pre-Jacobi-Jordan algebra built as a direct sum of a given pre-Jacobi-Jordan algebra (A, •) and its dual (A * , •), endowed with a non-degenerate symmetric bilinear form B, where • and • are the products defined on A and A * , respectively. Finally, after pre-Jacobi-Jordan algebras classification in dimension two, we thoroughly give some double constructions of pre-Jacobi-Jordan algebraic structures. Keywords. (pre)Jacobi-Jordan algebra, bimodule, matched pair, double construction MSC2010. 16T25, 05C25, 16S99, 16Z05.Preprint: ICMPA-UL/2019/12 Definition 2.4. [13] A vector space V is a module over a JJ algebra A, if there is a linear map (a representation) ρ : A → End(V ) such that ρ(x ⋄ y)(v) = −ρ(x)(ρ(y)v) − ρ(y)(ρ(x)v)(2.4)for any x, y ∈ A and v ∈ V .

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

“…Definition 3.4. [24] Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure (A * , ⋄) on the dual space A * of A and there is a mock-Lie algebra structure on the direct sum A ⊕ A * of the underlying vector spaces A and A * such that (A, •) and (A * , ⋄) are subalgebras and the natural non-degerenate symmetric bilinear form on A ⊕ A * given by Proof.…”

“…Proposition 3.4. [24] Let (A, •) be a mock-Lie algebra. Suppose that there is a mock-Lie algebra structure (A * , ⋄) on A * .…”

Representations and formal deformations of Hom-Leibniz algebras

“…This left pre-Jacobi-Jordan algebra is called a semi-direct product of A and V denoted by A ⋉ V. Remark 4.13. Our definition of representations of left pre-Jacobi-Jordan algebras given above is different of those given in [11]. Now, we can extend the notions of representations of pre-Jacobi-Jordan algebras in the Hom-case as follows: Definition 4.14.…”

“…As Pre-Lie algebras (also called left-symmetric algebras, quasi-associative algebras, Vinberg algebras and so on), left(resp. right)pre-Jacobi-Jordan algebras [11] are apparently first introduced in [3] under the name left skew-symmetric (resp. right skew-symmetric) algebras.…”

“…right skew-symmetric) algebras. The study of some relevant properties such as bimodules, matched pairs is considered for these algebras [11]. Moreover, a 2-dimensional classification and some double constructions of these algebras are given.…”