The structure of split regular Hom-Poisson algebras

Non-associative algebraic structures: classification and structure

“…In our joint work with Popov [168], we study the structure of split regular Hom-Leibniz 3-algebras of arbitrary dimension and over an arbitrary base field F. Split structures first appeared in the classical theory of (finite-dimensional) Lie algebras, but have been extended to more general settings like, for example, Leibniz algebras [53], Poisson algebras, Leibniz superalgebras, regular Hom-Lie algebras, regular Hom-Lie superalgebras, regular Hom-Lie color algebras, regular Hom-Poisson algebras [14], regular Hom-Leibniz algebras, regular BiHom-Lie algebras [54], and regular BiHom-Lie superalgebras, among many others. As for the study of split ternary structures, see [49] for Lie triple systems, twisted inner derivation triple systems, Lie 3-algebras [49], Leibniz 3-algebras [55], and for Leibniz triple systems.…”

Matching Hom-Setting of Rota-Baxter Algebras, Dendriform Algebras, and Pre-Lie Algebras

“…It is shown that a Homassociative algebra gives rise to a Hom-Lie algebra using the commutator. Since then, various Hom-analogues of some classical algebraic structures have been introduced and studied intensively, such as Hom-coalgebras, Hom-bialgebras and Hom-Hopf algebras [24,25], Hom-groups [26,27], Hom-Hopf modules [28], Hom-Lie superalgebras [29,30], generalize Hom-Lie algebras [31], and Hom-Poisson algebras [32].…”

Split regular Hom-Leibniz color 3-algebras

“…In the present paper we study the structure of split regular Hom-Leibniz 3-algebras of arbitrary dimension and over an arbitrary base field K. Split structures appeared first in the classical theory of (finite dimensional) Lie algebras but have been extended to more general settings like, for example, Leibniz algebras [13], Poisson algebras, Leibniz superalgebras [14], regular Hom-Lie algebras [3], regular Hom-Lie superalgebras [2], regular Hom-Lie color algebras [23], regular Hom-Poisson algebras [4], regular Hom-Leibniz algebras [24], regular BiHom-Lie algebras [16], regular BiHom-Lie superalgebras [43], among many others. As for the study of split ternary structures, see [10,11,12] for Lie triple systems, twisted inner derivation triple systems, Lie 3-algebras [12], Leibniz 3-algebras [17] and [21,22] for Leibniz triple systems.…”