On n-ary Generalization of BiHom-Lie Algebras and BiHom-Associative Algebras

Transposed BiHom-Poisson algebras

“…We only verify that Eq. (3.14) holds for (L, • ′ , [ , , ] ′ , α, β) and others can be obtained by [10,Claim 3.7] and [15,Theorem 1.12].…”

“…Roughly speaking, a BiHom-associative algebra (or Lie algebra) is an algebra (or Lie algebra) such that the associativity (or Jacobi condition) is twisted by two (commuting) endomorphisms, for details see [10], which can be seen as an extension of Hom-type algebra [13] arising in quasi-deformations of Lie algebras of vector fields. Now there are so many research related to BiHom-type algebras, see refs [5,11,12,[15][16][17][18][19][20][21][23][24][25][26][27][28]. In [21], the authors introduced the notion of BiHom-Poisson algebras and gave a necessary and sufficient condition under which BiHom-Novikov-Poisson algebras (which are twisted generalizations of Novikov-Poisson algebras [30] and Hom-Novikov-Poisson algebras [31]) give rise to BiHom-Poisson algebras.…”

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

“…Derivations, L-modules, L-comodules and Hom-Lie quasi-bialgebras have been considered in [29,30]. In [58], constructions of n-ary generalizations of BiHom-Lie algebras and BiHom-associative algebras have been considered. Generalized Derivations of n-BiHom-Lie algebras have been studied in [34].…”

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]…”