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Register a new userRepresentations and deformations of Hom-Lie-Yamaguti superalgebras
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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.
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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.
In this paper, first we give the cohomologies of an $n$-Hom-Lie algebra and introduce the notion of a derivation of an $n$-Hom-Lie algebra. We show that a derivation of an $n$-Hom-Lie algebra is a $1$-cocycle with the coefficient in the adjoint representation. We also give the formula of the dual representation of a representation of an $n$-Hom-Lie algebra. Then, we study $(n-1)$-order deformation of an $n$-Hom-Lie algebra. We introduce the notion of a Hom-Nijenhuis operator, which could generate a trivial $(n-1)$-order deformation of an $n$-Hom-Lie algebra. Finally, we introduce the notion of a generalized derivation of an $n$-Hom-Lie algebra, by which we can construct a new $n$-Hom-Lie algebra, which is called the generalized derivation extension of an $n$-Hom-Lie algebra.
In this paper, we define a new cohomology theory for multiplicative Hom-pre-Lie algebras which controls deformations of Hom-pre-Lie algebra structure. This new cohomology is a natural one considering the structure map $alpha$. We show this new cohomology is deformation cohomology of the Hom-pre-Lie algebra. We also develop cohomology and the associated deformation theory for Hom-pre-Lie algebras in the equivariant context.
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).
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.
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