LEFT-HOM-SYMMETRIC AND HOM-POISSON DIALGEBRAS

Abstract

The aim of this paper is to introduce left-Hom-symmetric dial- gebras (which contain left-Hom-symmetric algebras or Hom-preLie algebras and Hom-dialgebras as special cases) and Hom-Poisson dialgebras. We give some examples and some construction theorems by using the composition con- struction. We prove that the commutator bracket of any left-Hom-symmetric dialgebra provides Hom-Leibniz algebra. We also prove that bimodules over Hom-dialgebras are closed under twisting. Next, we show that bimodules over Hom-dendriform algebras D extend to bimodules over the left-Hom-symmetric algebra associated to D. Finally, we give some examples of Hom-Poisson dial- gebras and prove that the commutator bracket of any Hom-dialgebra structure map leads to Hom-Poisson dialgebra.