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In this paper, we introduce the notions of biderivations and linear commuting maps of Hom-Lie algebras and superalgebras. Then we compute biderivations of the q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebras by elementary and direct calculations. As an application, linear commuting maps on these algebras are characterized. Also, we introduce the notions of {alpha}-derivations and {alpha}-biderivations for Hom-Lie algebras and superal- gebras, and we establish a close relation between {alpha}-derivations and {alpha}-biderivations. As an illustration, we prove that the q-deformed W(2;2)-algebra, the q-deformed Witt algebra and superalgebra have no nontrivial {alpha}-biderivations. Finally, we present an example of Hom-Lie superalgebras with nontrivial {alpha}-super-derivations and biderivations.
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In this paper we attempt to investigate the super-biderivations of Lie superalgebras. Furthermore, we prove that all super-biderivations on the centerless super-Virasoro algebras are inner super-biderivations. Finally, we study the linear super commuting maps on the centerless super-Virasoro algebras.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
Let $(L, alpha)$ be a Hom-Lie-Yamaguti superalgebra. We first introduce the representation and cohomology theory of Hom-Lie-Yamaguti superalgebras. Furthermore, we introduce the notions of generalized derivations and representations of $(L, alpha)$ and present some properties. Finally, we investigate the deformations of $(L, alpha)$ by choosing some suitable cohomology.
After endowing with a 3-Lie-Rinehart structure on Hom 3-Lie algebras, we obtain a class of special Hom 3-Lie algebras, which have close relationships with representations of commutative associative algebras. We provide a special class of Hom 3-Lie-Rinehart algebras, called split regular Hom 3-Lie-Rinehart algebras, and we then characterize their structures by means of root systems and weight systems associated to a splitting Cartan subalgebra.
We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an $A$-split abelian extension and the equivalence classes of $A$-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras.
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