Center-symmetric Algebras and Bialgebras: Relevant Properties and Consequences

Abstract

Abstract. Lie admissible algebra structures, called center -symmetric algebras, are defined. Main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center -symmetric algebra on the direct sum of underlying vector space and a finite dimensional vector space. Then, the matched pair of center -symmetric algebras is established and related to the matched pair of sub-adjacent Lie algebras. Besides, Manin triple of center -symmetric algebras is defined and linked with their associated matched pairs. Further, center -symmetric bialgebras of center -symmetric algebras are investigated and discussed. Finally, a theorem yielding the equivalence between Manin triple of center -symmetric algebras, matched pairs of Lie algebras and center-symmetric bialgebras is provided.

Hom-center-symmetric algebras and bialgebras

“…In the particular case when α = id, the construction of the center-symmetric algebra structure on the semi-direct vector space A ⊕ V is given in [7]. By analogy to this work, we have: Proposition 3.2.…”

“…Sun and H. Li [17]. The center-symmetric algebras were investigated in [7], where their Lie-admissibility was also established, and their bimodules were constructed. Furthermore, their matched pairs were defined and linked to matched-pairs of Lie algebras and associated Manin triple.…”

Hom-left-symmetric color dialgebras, Hom-tridendriform color algebras and Yau’s twisting generalizations

“…In Hom-algebra structures, defining algebra identities are twisted by linear maps. Hom-algebras structures are very useful since Hom-algebra structures of a given type include their classical counterparts and open more possibilities for deformations, extensions of homology and cohomology structures and representations, Hom-coalgebra, Hom-bialgebras and Hom-Hopf algebras (see for example [1,[3][4][5][6][7][8][9][10][11]18,26,27,29,[33][34][35]50,[53][54][55]58,60,63] and references therein).…”

Hom-pre-Malcev and Hom-M-Dendriform algebras

“…Since the pioneering works [18,[29][30][31][32]47], Hom-algebra structures have developed in a popular broad area with increasing number of publications in various directions. Hom-algebra structures include their classical counterparts and open new broad possibilities for deformations, extensions to Hom-algebra structures of representations, homology, cohomology and formal deformations, Hommodules and hom-bimodules, Hom-Lie admissible Hom-coalgebras, Hom-coalgebras, Hom-bialgebras, Hom-Hopf algebras, L-modules, L-comodules and Hom-Lie quasibialgebras, n-ary generalizations of biHom-Lie algebras and biHom-associative algebras and generalized derivations, Rota-Baxter operators, Hom-dendriform color algebras, Rota-Baxter bisystems and covariant bialgebras, Rota-Baxter cosystems, coquasitriangular mixed bialgebras, coassociative Yang-Baxter pairs, coassociative Yang-Baxter equation and generalizations of Rota-Baxter systems and algebras, curved Ooperator systems and their connections with tridendriform systems and pre-Lie algebras, BiHom-algebras, BiHom-Frobenius algebras and double constructions, infinitesimal biHom-bialgebras and Hom-dendriform D-bialgebras, and category theory of Homalgebras [2,3,[5][6][7][8][9][10][11][12]15,16,[19][20][21][22][23][24]29,[32][33][34][37][38][39][40]42,45,[48][49][50]…”