Representations and cohomologies of Hom-Lie–Yamaguti algebras with applications
Tom 148 / 2017
Colloquium Mathematicum 148 (2017), 131-155 MSC: Primary 17D99; Secondary 18G60. DOI: 10.4064/cm6903-6-2016 Opublikowany online: 3 March 2017
The representation and cohomology theory of Hom-Lie–Yamaguti algebras are introduced. As an application, we study deformation and extension of Hom-Lie–Yamaguti algebras. It is proved that a 1-parameter infinitesimal deformation of a Hom-Lie–Yamaguti algebra $T$ corresponds to a Hom-Lie–Yamaguti algebra of deformation type and a $(2,3)$-cocycle of $T$ with coefficients in the adjoint representation. We also prove that abelian extensions of Hom-Lie–Yamaguti algebras are classified by the $(2,3)$-cohomology group.