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$\alpha$-type Hochschild cohomology of Hom-associative algebras and bialgebras
Benedikt Hurle,Abdenacer Makhlouf 대한수학회 2019 대한수학회지 Vol.56 No.6
In this paper we define a new type of cohomology for multiplicative Hom-associative algebras, which generalizes 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.
α-TYPE HOCHSCHILD COHOMOLOGY OF HOM-ASSOCIATIVE ALGEBRAS AND BIALGEBRAS
Hurle, Benedikt,Makhlouf, Abdenacer Korean Mathematical Society 2019 대한수학회지 Vol.56 No.6
In this paper we define a new type of cohomology for multiplicative Hom-associative algebras, which generalizes 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.
Ternary Distributive Structures and Quandles
Elhamdadi, Mohamed,Green, Matthew,Makhlouf, Abdenacer Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.1
We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from 3-Lie algebras are provided. We also describe ternary distributive algebraic structures coming from groups and give examples from vector spaces whose bases are elements of a finite ternary distributive set. We introduce a cohomology theory that is analogous to Hochschild cohomology and relate it to a formal deformation theory of these structures.