The present paper is devoted to study 2-local derivations on the Block-type Lie algebra which is an infinite-dimensional Lie algebra with some outer derivations. We prove that every 2-local derivation on the Block-type Lie algebra is a derivation.
In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which …
The quasi-filiform Lie algebras of nonzero rank are described. The classifications of filiform and quasi-filiform naturally graded algebras are corrected.
We study classical R-matrices D for Lie algebras L such that D is also a derivation of L. This yields derivation double Lie algebras (L,D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.
The concept of derivation for Lie-Yamaguti algebras is generalized in this paper. A quasi-derivation of an LY-algebra is embedded as derivation in a larger LY-algebra. The relationship between quasi-derivations and robustness of Lie-Yamaguti algebras has been studied.
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.
In the present paper we study local and 2-local derivations of locally finite split simple Lie algebras. Namely, we show that every local and 2-local derivation on such Lie algebra is a derivation.
$N$-derivation is the natural generalization of derivation and triple derivation. Let ${\cal L}$ be a finitely generated Lie algebra graded by a finite dimensional Cartan subalgebra. In this paper, a sufficient condition for Lie $N$-derivation algebra of ${\cal L}$ coinciding with Lie derivation algebra of ${\cal L}$ is given. As applications, any $N$-derivation of Schrödinger-Virasoro algebra, generalized Witt algebras, Kac-Moody algebras and their Borel subalgebras, is a derivation.
In this paper, we focus on the $(\si,\t)$-derivation theory of Lie conformal superalgebras. Firstly, we study the fundamental properties of conformal $(\si,\t)$-derivations. Secondly, we mainly research the interiors of conformal $G$-derivations. Finally, we discuss the relationships between the conformal $(\si,\t)$-derivations and some generalized conformal derivations of Lie conformal superalgebras.
We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras $\mathcal{L}$ with $\dim\mathcal{L}\geq 3$ which admit local derivations which are not derivations.
The present paper is devoted to study 2-local derivations on infinite-dimensional Lie algebras over a field of characteristic zero. We prove that all 2-local derivations on the Witt algebra as well as on the positive Witt algebra are (global)derivations, and give an example of infinite-dimensional Lie algebra with a 2-local derivation which is not a derivation.
We continue the algebraic study of almost inner derivations of Lie algebras over a field of characteristic zero and determine these derivations for free nilpotent Lie algebras, for almost abelian Lie algebras, for Lie algebras whose solvable radical is abelian and for several classes of filiform nilpotent Lie algebras. We find a family of $n$-dimensional characteristically nilpotent filiform Lie algebras $\mathfrak{f}_n$, for all $n\ge 13$, all of whose derivations are almost inner. Finally we compare the almost inner derivations of Lie algebras considered over two different fields $K\supseteq…
We study almost inner derivations of Lie algebras, which were introduced by Gordon and Wilson in their work on isospectral deformations of compact solvmanifolds. We compute all almost inner derivations for low-dimensional Lie algebras, and introduce the concept of fixed basis vectors for proving that all almost inner derivations are inner for $2$-step nilpotent Lie algebras determined by graphs, free $2$ and $3$-step nilpotent Lie algebras, free metabelian nilpotent Lie algebras on two generators, almost abelian Lie algebras and triangular Lie algebras. On the other hand we also exhibit famili…
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).
Motivated by the works of Wang [Y. Wang, \textit{Lie (Jordan) derivations of arbitrary triangular algebras,} Aequationes Mathematicae, \textbf{93} (2019), 1221-1229] and Moafian et al. [F. Moafian and H. R. Ebrahimi Vishki, \textit{Lie higher derivations on triangular algebras revisited,} Filomat, \textbf{30}(12) (2016), 3187-3194.], we shall study Lie higher derivations of arbitrary triangular algebras. In fact, it is shown that every Lie higher derivation on an arbitrary triangular algebra is proper, using the notion of maximal left (right) ring of quotients.
Jacobson proved that if a Lie algebra admits an invertible derivation, it must be nilpotent. He also suspected, though incorrectly, that the converse might be true: that every nilpotent Lie algebra has an invertible derivation. We prove that a Lie algebra is nilpotent if and only if it admits an invertible Leibniz-derivation. The proofs are elementary in nature and are based on well-known techniques.
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Lie conformal superalgebras. Firstly, we construct the semidirect product of a Lie conformal superalgebra and its conformal module, and study derivations of this semidirect product. Secondly, we develop cohomology theory of Lie conformal superalgebras and discuss some applications to the study of deformations of Lie conformal superalgebras. Finally, we introduce generalized derivations of Lie conformal superalgebras and study their properties.
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-adjoint matrices over a commutative $*$-ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. We also show that every local spatial derivation on the above Lie algebras is a derivation.
We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several characterizations, such as the existence of gradings by sixth roots of unity, or the existence of a nonsingular derivation whose inverse is again a derivation. We also obtain results on the existence of periodic prederivations. In this context we study a generalization of Engel-4-Lie algebras.
In this paper we describe the derivations of orthosymplectic Lie superalgebras over a superring. In particular, we derive sufficient conditions under which the derivations can be expressed as a semidirect product of inner and outer derivations. We then present some examples for which these conditions hold.
We study $δ$-derivations -- a construction simultaneously generalizing derivations and centroid. First, we compute $δ$-derivations of current Lie algebras and of modular Zassenhaus algebra. This enables us to provide examples of Lie algebras having 1/2-derivations which are divisors of zero, thus answering negatively a question of Filippov. Second, we note that $δ$-derivations allow, in some circumstances, to construct examples of non-semigroup gradings of Lie algebras, in addition to the recent ones discovered by Elduque. Third, we note that utilizing the construction of the Grassmann envelop…
In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation.
We provide that any Jordan derivation from the block upper triangular matrix algebra $\T = \T(n_{1},n_{2}, \cdots, n_{k})\subseteq M_{n}(\mathbb{\C})$ into a $2$-torsion free unital $\T$-bimodule is the sum of a derivation and an antiderivation.
For a Lie algebra $\mathfrak g$ related to a quantum torus, we compute its automorphisms, derivations and universal central extension. This Lie algebra $\mathfrak g$ is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.
Motivated by the Cheung's elaborate work [Linear Multilinear Algebra, 51 (2003), 299-310], we investigate the construction of a Lie derivation on a generalized matrix algebra and apply it to give a characterization for such a Lie derivation to be proper. Our approach not only provides a direct proof for some known results in the theory, but also it presents several sufficient conditions assuring the properness of Lie derivations on certain generalized matrix algebras.
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce $α^k$-derivations of multiplicative Hom-Lie conformal algebras and study their properties.
It is well known that n-Hom Lie superalgebras are certain generalizations of n-Lie algebras. This paper is devoted to investigate the generalized derivations of multiplicative n-Hom Lie superalgebras. We generalize the main results of Leger and Luks to the case of multiplicative n-Hom Lie superalgebras. Firstly, we review some concepts associated with a multiplicative n-Hom Lie superalgebra $N$. Furthermore, we give the definitions of the generalized derivations, quasiderivations, center derivations, centroids and quasicentroids. Obviously, we have the following tower $ZDer(N)\subseteq Der(N)\…
We study solvable Lie algebras in prime characteristic $p$ that admit non-singular derivations. We show that Jacobson's Theorem remains true if the quotients of the derived series have dimension less than~$p$. We also study the structure of Lie algebras with non-singular derivations in which the derived subalgebra is abelian and has codimension~one. The paper presents some new examples of solvable, but not nilpotent, Lie algebras of derived length~3 with non-singular derivations.
In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether the algebra admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nilpotent dimensionally nilpotent Lie triple algebras. We show that when $n=2p+1$ the adapted basis coincides with the canonical basis of the gametic algebra $G(2p+2,2)$ or this one obviously associated to a pseudo-idempotent and if $n=2p$ then the algebra is either one of the precedent case or a conservative Bernstein algebra.
The aim of this article is to discuss the $n$-derivation algebras of Lie color algebras. It is proved that, if the base ring contains $\frac{1}{n-1}$, $L$ is a perfect Lie color algebra with zero center, then every triple derivation of $L$ is a derivation, and every $n$-derivation of the derivation algebra $nDer(L))$ is an inner derivation.
In the present paper we prove the existence outer derivations finite-dimensional solvable Lie algebras with nilradical of maximal rank and complementary subspace to nilardical of dimension less than rank of the nilradical.
In this paper, we study triple derivations and triple homomorphisms of perfect Lie superalgebras over a commutative ring $R$. It is proved that, if the base ring contains $\frac{1}{2}$, $L$ is a perfect Lie superalgebra with zero center, then every triple derivation of $L$ is a derivation, and every triple derivation of the derivation algebra $ Der (L)$ is an inner derivation. Let $L,~L^{'}$ be Lie superalgebras over a commutative ring $R$, the notion of triple homomorphism from $L$ to $L^{'}$ is introduced. We proved that, under certain assumptions, homomorphisms, anti-homomorphisms, and sums…
In this paper, we give some basic properties of the generalized derivation algebra ${\rm GDer}(L)$ of a Hom-Lie superalgebra $L$. In particular, we prove that ${\rm GDer}(L) = {\rm QDer}(L) + {\rm QC}(L)$, the sum of the quasiderivation algebra and the quasicentroid. We also prove that ${\rm QDer}(L)$ can be embedded as derivations in a larger Hom-Lie superalgebra.
In this paper we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers $\mathbb{N}$, describe the lattice of its ideals for arbitrary field $K$ and study its derivations over any commutative, unital ring $R$.
Let $P$ be an arbitrary partially ordered set, $R$ a commutative ring with identity and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. Under some natural assumption on $R$, we prove that each Lie-type derivation of $FI(P,R)$ is proper.
In this paper, we present some basic properties concerning the derivation algebra ${\rm Der}(T)$, the quasiderivation algebra ${\rm QDer}(T)$ and the generalized derivation algebra ${\rm GDer}(T)$ of a Lie triple system $T$, with the relationship ${\rm Der}(T)\subseteq {\rm QDer}(T)\subseteq {\rm GDer}(T)\subseteq {\rm End}(T)$. Furthermore, we completely determine those Lie triple systems $T$ with condition ${\rm QDer}(T)={\rm End}(T)$. We also show that the quasiderivations of $T$ can be embedded as derivations in a larger Lie triple system.
A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$, with the known cohomology of $n$-Lie algebras. The abelian extensions and infinitesimal deformations of $n$-Lie algebras, on the other hand, are shown to be characterized by the usual cohomology of $n$-Lie algebras. Furthermore, the Hochschild-Serre spectral sequence of the Lie algebra cohomology is upgraded to the level of $n$-Lie algebras, and is applied …
Let $A$ be a von Neumann algebra with no central summands of type $I_1$. We will show that every nonlinear Lie $n$-derivation on $A$ is of the standard form, i.e. it can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each $(n-1)$-th commutator of $A$.
In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.
Let $K$ be a field and $Γ$ a finite quiver without oriented cycles. Let $Λ$ be the path algebra $K(Γ, ρ)$ and let $\mathscr{D}(Λ)$ be the dual extension of $Λ$. In this paper, we prove that each Lie derivation of $\mathscr{D}(Λ)$ is of the standard form.
We study almost inner derivations of $2$-step nilpotent Lie algebras of genus $2$, i.e., having a $2$-dimensional commutator ideal, using matrix pencils. In particular we determine all almost inner derivations of such algebras in terms of minimal indices and elementary divisors over an arbitrary algebraically closed field of characteristic not $2$ and over the real numbers.
Let $X$ be a locally finite preordered set, $\mathcal R$ a commutative ring with identity and $I(X,\mathcal R)$ the incidence algebra of $X$ over $\mathcal R$. In this note we prove that each Lie derivation of $I(X,\mathcal R)$ is proper, provided that $\mathcal R$ is $2$-torsion free.
In this paper, we introduce the notion of a BiHom-Lie conformal algebra and develop its cohomology theory. Also, we discuss some applications to the study of deformations of regular BiHom-Lie conformal algebras. Finally, we introduce derivations of multiplicative BiHom-Lie conformal algebras and study their properties.
Local and 2-local $\frac{1}2$-derivations of infinite-dimensional Lie algebras
V.I.Romanovskiy Institute of Mathematics Uzbekistan Academy of Sciences（乌兹别克斯坦科学院V.I.罗曼诺夫斯基数学研究所）；National University of Uzbekistan（乌兹别克斯坦国立大学）；Urgench State University（urganch 州立大学）
In this paper, we consider a 3-Lie algebra with a derivation (called a 3-LieDer pair). We define cohomology for a 3-LieDer pair with coefficients in a representation. We use this cohomology to study deformations and abelian extensions of 3-LieDer pairs. We give the notion of a 3-Lie2Der pair, which can be viewed as the categorification of a 3-LieDer pair. We show that skeletal 3-Lie2Der pairs are classified by triples given by 3-LieDer pairs, representations and 3-cocycles. We define crossed modules of 3-LieDer pairs and show that there exists a one-to-one correspondence between strict 3-Lie2D…
Building on work of Gerstenhaber, we show that the space of integrable derivations on an Artin algebra $A$ forms a Lie algebra, and a restricted Lie algebra if $A$ contains a field of characteristic $p$. We deduce that the space of integrable classes in $\HH^1(A)$ forms a (restricted) Lie algebra that is invariant under derived equivalences, and under stable equivalences of Morita type between self-injective algebras. We also provide negative answers to questions about integrable derivations posed by Linckelmann and by Farkas, Geiss and Marcos.
We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a periodic derivation of order $n$. We find an optimal bound on the nilpotency class in characteristic $p$ if $p$ does not divide a certain invariant $ρ_n$. We give a new description of the set $\mathcal{N}_p$ of positive integers $n$, introduced by Shalev, which arise as the order of a periodic derivation of a finite-dimensional non-nilpotent Lie algebra in c…
In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a byproduct, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad-locally finite element.
Let $T$ be a $δ$-Jordan Lie supertriple system. We first introduce the notions of generalized derivations and representations of $T$ and present some properties. Also, we study the low dimension cohomology and the coboundary operator of $T$, and then we investigate the deformations and Nijenhuis operators of $T$ by choosing some suitable cohomology.
