Hom-center-symmetric algebras and bialgebras

Abstract

In this work, the hom-center-symmetric algebras are constructed and discussed. Their bimodules, dual bimodules and matched pairs are defined. The relation between the dual bimodules of hom-center-symmetric algebras and the matched pairs of hom-Lie algebras is established. Furthermore, the Manin triple of hom-center-symmetric algebras is given. Finally, a theorem linking the matched pairs of hom-center-symmetric algebras, the hom-center-symmetric bialgebras and the matched pairs of sub-adjacent hom-Lie algebras is provided.

Constructions of BiHom-X algebras and bimodules of some BiHom-dialgebras

“…Let (D, ⊣, ⊢, α, β) be a BiHom-associative dialgebra, then ( 15) and ( 16) are satisőed. Since the products ⊣ and ⊢ are associative with the condition (23), the equalities ( 17) and ( 18) are established.…”

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]…”

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 [1,3,4,5,6,7,8,9,10,16,24,25,27,31,32,33,49,52,53,54,57,59,62] and references therein).…”