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Register a new userA generalization on derivations of Lie algebras
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We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
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In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, { , , , }_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $Altimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equation in the 3-Lie algebra $Altimes_{ad^*}A^*$, a class of 3-pre-Lie algebras, and eight and ten dimensional local cocycle 3-Lie bialgebras are constructed.
For any $n$-dimensional 3-Lie algebra $A$ over a field of characteristic zero with an involutive derivation $D$, we investigate the structure of the 3-Lie algebra $B_1=Altimes_{ad^*} A^* $ associated with the coadjoint representation $(A^*, ad^*)$. We then discuss the structure of the dual 3-Lie algebra $B_2$ of the local cocycle 3-Lie bialgebra $(Altimes_{ad^*} A^*, Delta)$. By means of the involutive derivation $D$, we construct the $4n$-dimensional Manin triple $(B_1oplus B_2,$ $ [ cdot, cdot, cdot]_1,$ $ [ cdot, cdot, cdot]_2,$ $ B_1, B_2)$ of 3-Lie algebras, and provide concrete multiplication in a special basis $Pi_1cupPi_2$. We also construct a sixteen dimensional Manin triple $(B, [ cdot, cdot, cdot])$ with $dim B^1=12$ using an involutive derivation on a four dimensional 3-Lie algebra $A$ with $dim A^1=2$.
We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.
Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.
In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $sigma$-derivations. We show that $sigma$-twisted Jacobi type identity holds for generators of such deformations. For the $sigma$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also corresponding q-deformed Jacobi identities.
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