\(T^*\)-extensions and abelian extensions of Hom-Lie color algebras. (English) Zbl 1439.17024
Summary: We study hom-Nijenhuis operators, \(T^*\)-extensions and abelian extensions of hom-Lie color algebras. We show that the infinitesimal deformation generated by a hom-Nijenhuis operator is trivial. Many properties of a hom-Lie color algebra can be lifted to its \(T^*\)-extensions such as nilpotency, solvability and decomposition. It is proved that every finite-dimensional nilpotent quadratic hom-Lie color algebra over an algebraically closed field of characteristic not 2 is isometric to a \(T^*\)-extension of a nilpotent Lie color algebra. Moreover, we introduce abelian extensions of hom-Lie color algebras and show that there is a representation and a 2-cocycle, associated to any abelian extension.
MSC:
| 17B61 | Hom-Lie and related algebras |
| 17B30 | Solvable, nilpotent (super)algebras |
| 17B75 | Color Lie (super)algebras |
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