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 alpha-type Hochschild cohomology of Hom-associative algebras and bialgebras J. Korean Math. Soc.Published online April 10, 2019 Benedikt HURLE and Abdenacer MAKHLOUF Université de Haute Alsace, France Abstract : In this paper we define a new type cohomology for multiplicative Hom-associative algebras, which generalize Hom-type Hochschild cohomology and fits with deformations of Hom-associative algebras including the deformation of the structure map alpha. Moreover, we provide various observations and similarly a new type cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras and fitting with formal deformations including deformations of the structure map. Keywords : Cohomology, Hom-associative algebra, deformation, Hom-bialgebra, $L_\infty$-structure. MSC numbers : 16W55, 16S99, 16S80,16T10 Full-Text :