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AbstractIn this paper we prove that the variety Cm(g) of commuting m-tuples of elements of a simple Lie algebra g is usually reducible. More precisely, one of the main results of this paper shows that if m=3 and g is of types B,C,E7,E8,F4,G2 or Dl with l≥10, then Cm(g) is reducible. Moreover, if m≥4 and g is a simple Lie algebra different from sl2,sl3, then Cm(g) is reducible. In order to obtain these results, we need a more general result, which is another main result of this paper. It states that, under some mild assumptions, Cm(g) is reducible whenever Cm(l) is reducible for a Levi subalgebra l of g. [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classify some special types of homotopy pre-Leibniz algebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper we define the H-module Lie algebra of quotients for an H-semiprime Lie algebra L, where H is a cocommutative Hopf algebra and we compute the maximal H-module Lie algebra of quotients of L, say QmH(L). As an application, we compute the Gelfand-Kirillov dimension (GK) of QmH(L) showing it equals the GK dimension of L. [ABSTRACT FROM AUTHOR]

Leibniz algebras are certain generalizations of Lie algebras. Motivated by the concept of subinvariance and ascendancy in group theory, Schenkman studied properties of subinvariant subalgebras of a Lie algebra. Kawamoto studied ascendency in Lie algebras. In this paper we define subinvariance and asendency in Leibniz algebras and study their properties. It is shown that the signature results on subinvariance and ascendency in Lie algebras have analogs for Leibniz algebras. [ABSTRACT FROM AUTHOR]

This paper is concerned with generalizing the results for Lie CT-algebras to Leibniz algebras. In some cases our results give a generalization even for the case of a Lie algebra. Results on A-algebras are used to show every Leibniz CT-algebra over an algebraically closed field of characteristic different from 2,3 is solvable or is isomorphic to s l 2 (F) . A characterization is then given for solvable Leibniz CT-algebras. It is also shown that the class of solvable Leibniz CT-algebras is factor closed. [ABSTRACT FROM AUTHOR]

Let m, n be integers with m > n . Denote M m , n by the set of all m × n matrices over the field F of characteristic zero. Let I be an n × m matrix with (i , i) -position 1 for any 1 ≤ i ≤ n , and 0 in other position. Define a bracket [ A , B ] = AIB − BIA , where A , B ∈ M m , n . Then M m , n with this bracket is a Lie algebra, called non-square linear Lie algebra, denoted by gl (m × n , F) . In this paper, all derivations and biderivations of gl (m × n , F) are determined. As applications of biderivations, the linear commuting maps and commutative post-Lie algebra structures on gl (m × n , F) are given. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra s c h n : = s l 2 ⋉ h n is the semi-direct product of the Lie algebra s l 2 with the n-th Heisenberg Lie algebra h n . In this paper, we classify the s c h n -modules (n ≥ 2) that are free of rank 1 when restricted to the Cartan subalgebra (modulo center). Furthermore, the simplicities of these modules are determined. [ABSTRACT FROM AUTHOR]

In this paper, we classify eight-dimensional non-solvable Lie algebras that support a symplectic structure. As well as a complete classification is given, up to symplectomorphism, of eight-dimensional symplectic non-solvable Lie algebras. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra is the semidirect product of the special linear Lie algebra s l 2 and the Heisenberg algebra h n . In this paper, we describe explicitly the structure of the derivation algebra and automorphism group of the n-th Schrödinger algebra. Especially, the automorphism group of the Schrödinger algebra is obtained. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of local derivations and automorphisms of nilpotent Lie algebras. Namely, we proved that nilpotent Lie algebras with indices of nilpotency 3 and 4 admit a local derivation (local automorphism) which is not a derivation (automorphism). Further, it is presented a sufficient condition under which a nilpotent Lie algebra admits a local derivation which is not a derivation. With the same condition, it is proved the existence of pure local automorphism on a nilpotent Lie algebra. Finally, we present an n-dimensional nilpotent non-associative algebra for which the space of local derivations coincides with the space of derivations. [ABSTRACT FROM AUTHOR]

For the parameter b ∈ C , denote the semi-direct product of the Witt algebra and the loop Schrödinger algebra as S W (b) . In this paper, we investigate the skew-symmetric biderivations of this Lie algebra. As an application, we give the explicit form of each linear commuting map on S W (b) . It turns out that on S W (b) we have only inner biderivations and standard linear commuting maps except in the case b = − 1 2 . [ABSTRACT FROM AUTHOR]

The present paper is devoted to the description of local derivations on solvable Lie algebras of maximal rank. Namely, we consider a solvable Lie algebra of the form R = Q ⊕ N , where Q is the maximal torus subalgebra of R , N is the nilradical of R and dim Q = dim N / N 2. We prove that any local derivation of such solvable Lie algebra R is a derivation. Further, we present two examples of solvable Lie algebras which satisfy the condition dim Q < dim N / N 2 and the first algebra admit a local derivation which is not a derivation, while for the second algebra we prove that any local derivation is a derivation. We also apply the main result of the paper to the description of local derivations on so-called standard Borel subalgebras of complex simple Lie algebras. [ABSTRACT FROM AUTHOR]

The notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete. [ABSTRACT FROM AUTHOR]

Assume that R be a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent P. In this paper, it is shown that under certain conditions, ϕ ([ A , B ]) = [ ϕ (A) , B ] where ϕ an additive map on R and AB = P if and only if it has the form ϕ (A) = λ A + h (A) , where h is an additive map into its center vanishing at commutators [ A , B ] with A B = P and λ ∈ Z (R) . An application of our results we characterize generalized Lie derivations on R. Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

The weight modules of the Lie algebra s l (2 , C) are well known. In the first part of this paper we deal with a realization of weight modules of s l (2 , C) in the space t ν C [ t , t − 1 ] , ν ∈ R , where C [ t , t − 1 ] is the algebra of Laurent polynomials. In the second part, we consider the Hom-Lie algebra s l q (2 , C) of Jackson where q ≠ 0 , 1 . The q-analogue of the above realization in the space t ν C (q) [ t , t − 1 ] , ν ∈ R is considered. We obtain two kinds of q-modules. The regular q-modules which have limits the modules obtained in the classical realization when q goes to 1. The other q-modules have no limits when q goes to 1 and they are called singular modules. [ABSTRACT FROM AUTHOR]

Given a simple undirected graph, one can construct from it a c-step nilpotent Lie algebra for every c ≥ 2 and over any field K, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of K-forms in these real and complex Lie algebras for any subfield K ⊂ C from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions. [ABSTRACT FROM AUTHOR]

In this paper we study the Lie algebras of derivations of two-step nilpotent algebras. We obtain a class of Lie algebras with trivial center and abelian ideal of inner derivations. Among these, the relations between the complex and the real case of the indecomposable Heisenberg Leibniz algebras are thoroughly described. Finally we show that every almost inner derivation of a complex nilpotent Leibniz algebra with one-dimensional commutator ideal, with three exceptions, is an inner derivation. [ABSTRACT FROM AUTHOR]

In this paper, we answer a question asked by Kaiming Zhao at the mini course "Simple representations of the Virasoro algebra" at Xiamen mathematical center in the fall term of 2021. More precisely, we prove that if one special element in the Lie algebra g (including the Virasoro algebra, the Heisenberg-Virasoro algebra and affine Virasoro algebras) acts locally nilpotently on an irreducible module V, then V is a restricted g -module. [ABSTRACT FROM AUTHOR]

Let (G , Ω) be a symplectic Lie group, i.e., a Lie group endowed with a left invariant symplectic form. If g is the Lie algebra of G then we call (g , ω = Ω (e)) a symplectic Lie algebra. The product • on g defined by 3 ω (x • y , z) = ω ([ x , y ] , z) + ω ([ x , z ] , y) extends to a left invariant connection ∇ on G which is torsion free and symplectic ( ∇ Ω = 0) . When ∇ has vanishing curvature, we call (G , Ω) a flat symplectic Lie group and (g , ω) a flat symplectic Lie algebra. In this paper, we study flat symplectic Lie groups. We start by showing that the derived ideal of a flat symplectic Lie algebra is degenerate with respect to ω. We show that a flat symplectic Lie group must be nilpotent with degenerate center. This implies that the connection ∇ of a flat symplectic Lie group is always complete. We prove that the flat symplectic double extension process can be applied to characterize all flat symplectic Lie algebras. More precisely, we show that every flat symplectic Lie algebra is obtained by a sequence of flat symplectic double extension of flat symplectic Lie algebras starting from {0}. As examples in low dimensions, we classify all flat symplectic Lie algebras of dimension ≤ 6 . [ABSTRACT FROM AUTHOR]

Let R be an associative ring and let s ≥ 1 be a fixed integer. An additive map h on R is called a homoderivation if h (x y) = h (x) h (y) + h (x) y + x h (y) holds for all x , y ∈ R. In 1978, Herstein proved that a prime ring R of char (R) ≠ 2 is commutative if there is a nonzero derivation d of R such that [ d (x) , d (y) ] = 0 for all x , y ∈ R . The main objective of this paper is to prove the above mentioned result for homoderivations with nilpotency 's' in prime rings. Moreover, we prove that if a prime ring admits homoderivations h1 and h2 such that h 1 ° h 2 = h 2 ° h 1 and h 1 s 1 ° h 2 s 2 = 0 for positive integers s1 and s2, then at least one of the homoderivations must be nilpotent. [ABSTRACT FROM AUTHOR]

Recently, Lie bialgebra structures of the derivation Lie algebra over quantum torus were determined. In this paper, we use the general method of quantization by a Drinfel'd twist to quantize these algebras with their Lie bialgebra structures and present the explicit formulae. [ABSTRACT FROM AUTHOR]

In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]

This paper aims to introduce the concept of nilpotency and capability in multiplicative Lie algebras. Also, we see the existence of covers of a multiplicative Lie algebra and thoroughly examine their relationships with capable and perfect multiplicative Lie algebras. [ABSTRACT FROM AUTHOR]

Motivated by recent work on compatible mock-Lie algebras such that any linear combination of the two mock-Lie structures is a mock-Lie structure, in this paper we introduce compatible mock-Lie bialgebras as an analogue of compatible Lie bialgebras. They can also be regarded as a "compatible version" of mock-Lie bialgebras. Constructions of compatible mock-Lie bialgebras are presented via Manin triples and matched pairs. [ABSTRACT FROM AUTHOR]

A twisted affine Lie superalgebra is either a twisted affine Lie algebra or of one of the types X = A (2 m − 1 , 2 n − 1) (2) ( m , n ≠ 0 , (m , n) ≠ (1 , 1) ), A (2 m , 2 n) (4) , A (2 m , 2 n − 1) (2) or D (m + 1 , n) (2) ( m ≥ 0 , n > 0 ). It is known that irreducible integrable highest weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0. In this paper, we show that nonzero level irreducible integrable finite weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0. [ABSTRACT FROM AUTHOR]

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that 2-term homotopy Lie triple systems is equivalent to Lie triple 2-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple 2-systems. We show that skeletal Lie triple 2-systems can be classified the third cohomology group, and strict Lie triple 2-systems are equivalent to crossed modules of Lie triple systems. [ABSTRACT FROM AUTHOR]

k-symplectic manifolds are a convenient framework to study classical field theories and they are a generalization of polarized symplectic manifolds. This paper focus on the existence and the properties of left invariant k-symplectic structures on Lie groups having a bi-invariant pseudo-Riemannian metric. We show that compact semi-simple Lie groups and a large class of Lie groups having a bi-invariant pseudo-Riemannian metric does not carry any left invariant k-symplectic structure. This class contains the oscillator Lie groups which are the only solvable non abelian Lie groups having a bi-invariant Lorentzian metric. However, we built a natural left invariant n-symplectic structure on SL (n , R) . Moreover, up to dimension 6, only three connected and simply connected Lie groups have a bi-invariant indecomposable pseudo-Riemannian metric and a left invariant k-symplectic structure, namely, the universal covering of SL (2 , R) with a 2-symplectic structure, the universal covering of the Lorentz group SO (3 , 1) with a 2-symplectic structure, and a 2-step nilpotent 6-dimensional connected and simply connected Lie group with both a 1-symplectic structure and a 2-symplectic structure. [ABSTRACT FROM AUTHOR]

In this paper we describe the central extensions of some nilpotent Leibniz algebras and one-dimensional central extensions (up to isomorphism) of all naturally graded 2-filiform Leibniz algebras over the field of complex numbers. It is well-know that 2-filiform Leibniz algebras which are not naturally graded have not yet studied. There are such kind of algebras in the algebras obtained along this article. Also in this work we describe some subclasses of 6-dimensional nilpotent Leibniz algebras with condition quotient algebra by two-dimensional center of the algebra is isomorphic to the four-dimensional naturally graded 2-filiform nonsplit Leibniz algebra. [ABSTRACT FROM AUTHOR]

In this paper, we use Gröbner–Shirshov bases method for Leibniz (super)algebras to present a complete characterization of extensions of a Leibniz (super)algebra by another Leibniz (super)algebra, where the former is presented by generators and relations. [ABSTRACT FROM AUTHOR]

Let G be a classical group of dimension d and let a = (a 1 , ... , a d) be differential indeterminates over a differential field F of characteristic zero with algebraically closed field of constants C. Further let A (a) be a generic element in the Lie algebra g (F 〈 a 〉) of G obtained from parametrizing a basis of g with the indeterminates a. It is known (cf. [5]) that the differential Galois group of y ′ = A (a) y over F 〈 a 〉 is G (C) . In this paper we construct a differential field extension L of F 〈 a 〉 such that the field of constants of L is C, the differential Galois group of y ′ = A (a) y over L is still the full group G (C) and A (a) is gauge equivalent over L to a matrix in normal form which we introduced in [13]. We also consider specializations of the coefficients of A (a) . [ABSTRACT FROM AUTHOR]

In this paper, we study non-abelian extensions of pre-Lie algebras. First we introduce the non-abelian cohomology for pre-Lie algebras by which we classify non-abelian extensions of pre-Lie algebras. Then we construct the bimultipliers for pre-Lie algebras, which naturally gives rise to a strict Lie 2-algebra. We show that there is a one-to-one correspondence between isomorphic classes of non-abelian extensions of pre-Lie algebras and homotopy classes of homomorphisms from the subadjacent Lie algebra to the strict Lie 2-algebra obtained from the bimultipliers. Finally we classify non-abelian extensions of pre-Lie algebras by Maurer-Cartan elements of a differential graded Lie algebra. [ABSTRACT FROM AUTHOR]

In this paper, we give a method to compute the Gelfand-Kirillov dimensions of some polynomial-type weight modules. These modules are infinite-dimensional irreducible 픬(n,ℂ)-modules and 픭(2n,ℂ)-modules that appeared in the ℤ-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules. [ABSTRACT FROM AUTHOR]

The n-th Schrödinger algebra s c h n : = s l 2 ⋉ h n is the semi-direct of the Lie algebra s l 2 with the n-th Heisenberg Lie algebra h n . In this paper, all derivations and biderivations of the n-th Schrödinger algebra are determined. As applications, all linear commuting maps and commutative post-Lie algebra structures on s c h n are obtained. [ABSTRACT FROM AUTHOR]

In this paper, first we introduce the notion of a Reynolds operator on an n-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an n-Lie algebra. We give the cohomology theory of Reynolds operators on an n-Lie algebra and study infinitesimal deformations of Reynolds operators using the second cohomology group. Then we introduce the notion of NS-n-Lie algebras, which are generalizations of both n-Lie algebras and n-pre-Lie algebras. We show that an NS-n-Lie algebra gives rise to an n-Lie algebra together with a representation on itself. Reynolds operators and Nijenhuis operators on an n-Lie algebra naturally induce NS-n-Lie algebra structures. Finally, we construct Reynolds (n + 1) -Lie algebras and Reynolds 3-Lie algebras from Reynolds n-Lie algebras and Reynolds commutative associative algebras respectively. [ABSTRACT FROM AUTHOR]

Suppose the ground field is algebraically closed and of characteristic different from 2. In this paper, we described the intrinsic connections among linear super-commuting maps, super-biderivations and centroids for Lie superalgebras satisfying certain assumptions. This is a generalization of the results of Bresar and Zhao on Lie algebras. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the structure of all non-isomorphic finite-dimensional nilpotent Lie algebras L of class 3 with derived subalgebra of dimension 3 when dim Z (L) = 1. [ABSTRACT FROM AUTHOR]

A linear map φ on a Lie algebra g is called a triple derivation if φ ([ [ x , y ] , z ]) = [ [ φ (x) , y ] , z ] + [ [ x , φ (y) ] , z ] + [ [ x , y ] , φ (z) ] for all x , y , z ∈ g. Let g be a Kac-Moody algebra over an algebraically closed field of characteristic 0, and p an arbitrary parabolic subalgebra of g. In this paper, we prove that any triple derivation of p is a derivation, which generalizes the corresponding existing result for finite-dimensional simple Lie algebra. [ABSTRACT FROM AUTHOR]

In this paper, we study the classification of finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. We present an effective method to classify (n + 1) -dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given. In addition, the problem of classifying (n + 2) -dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation. [ABSTRACT FROM AUTHOR]

In the present paper, we study the notion of the Schur multiplier M (L) of an n-Lie superalgebra L = L 0 ⊕ L 1 and prove that dim M (L) ≤ ∑ i = 0 n ( m i ) L (n − i , k) , where dim L 0 = m , dim L 1 = k , L (0 , k) = 1 and L (t , k) = ∑ j = 1 t ( t − 1 j − 1 ) ( k j ) , for 1 ≤ t ≤ n. Moreover, we obtain an upper bound for the dimension of M (L) in which L is a nilpotent n-Lie superalgebra with one-dimensional derived superalgebra. It is also provided several inequalities on dim M (L) as well as an n-Lie superalgebra analogue of the converse of Schur's theorem. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show that there are Hom-Lie algebra structures on s l 2 (F) ⊕ F D , where D is a special type of generalized derivation of s l 2 (F) , and F is an algebraically closed field of characteristic zero. We study the representation theory of Hom-Lie algebras within the appropriate category and prove that any finite dimensional representation of a Hom-Lie algebra of the form s l 2 (F) ⊕ F D is completely reducible, analogously to the well-known theorem of Weyl from the classical Lie theory. We apply this result to characterize the non-solvable Lie algebras of this type having an invertible generalized derivation D. [ABSTRACT FROM AUTHOR]

Let be the ring of Laurent polynomials with commuting variables, and the Lie algebra of divergence-zero derivations on A. In this paper, we determine the set of derivations and the automorphism group for the derived subalgebra L of. [ABSTRACT FROM AUTHOR]

In this paper, we study partial group actions on Lie algebras. We describe the structure of the inverse semigroup of all partial automorphisms (isomorphisms between ideals) of a finite-dimensional reductive Lie algebra. Also, we show that every partial group action on a finite-dimensional semisimple Lie algebra admits a globalization, unique up to isomorphism. As a consequence, we obtain that the globalization problem for partial group actions on reductive Lie algebra is equivalent to the globalization problem on its center. [ABSTRACT FROM AUTHOR]

In a recent paper by Zhao and the author, the Lie algebras A[D] = A ⊗ F[D] of Weyl type were defined and studied, where A is a commutative associative algebra with an identity element over a field F of any characteristic, and F[D] is the polynomial algebra of a commutative derivation subalgebra D of A. In the present paper, the 2-cocycles of a class of the above Lie algebras A[D] (which are called the Lie algebras of generalized differential operators in the present paper), with F being a field of characteristic 0, are determined. Among all the 2-cocycles, there is a special one which seems interesting. Using this 2-cocycle, the central extension of the Lie algebra is defined. [ABSTRACT FROM AUTHOR]

It is well known that super derivation of a Lie superalgebra is certain generalization of derivation of a Lie algebra. This paper is devoted to investigate the structure and dimension of super derivation algebra Der(G) of G where G is a direct sum of two finite dimensional Lie superalgebras L and K having no non-trivial common direct factor. We also introduce some of its sub super algebras. Moreover, we create a condition which shows the isomorphism between super derivation of direct sum and direct sum of super derivations of two Lie superalgebras. Later on, we take G as a semidirect sum of two Lie superalgebras and obtain the structure of Der(G : K) which is a sub super algebra of Der(G) that contains those super derivations mapping K to itself. Finally, we give some conditions under which Der(G : K) is also a semidirect sum. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of non-abelian tensor product of a given precrossed module by one of its ideals. We use this concept to study the second homology and the nilpotent multiplier of a precrossed module. We also describe the Ganea term for the c-nilpotent multiplier and the universal central extension of precrossed modules in term of the non-abelian tensor product. [ABSTRACT FROM AUTHOR]

We introduce and study a new class of algebras, which we name quantum generalized Heisenberg algebras (qGHA), including both the so-called generalized Heisenberg algebras and the generalized down-up algebras, but allowing more parameters of freedom, so as to encompass a wider range of applications and provide a common framework for several previously studied classes of algebras. In particular, our class includes the enveloping algebras of the Lie algebra s l 2 and of the 3-dimensional Heisenberg Lie algebra, as well as its q-deformation, neither of which can be realized as a generalized Heisenberg algebra. This paper focuses mostly on the classification of finite-dimensional irreducible representations of qGHA, a study which reveals their rich structure. Although these algebras are not in general noetherian, their representations still retain a Lie-theoretic flavor. We work over a field of arbitrary characteristic and our results are presented in a characteristic-free fashion. [ABSTRACT FROM AUTHOR]

In this paper, we begin to study the subalgebra lattice of a Leibniz algebra. In particular, we deal with Leibniz algebras whose subalgebra lattice is modular, upper semi-modular, lower semi-modular, distributive, or dually atomistic. The fact that a non-Lie Leibniz algebra has fewer one-dimensional subalgebras in general results in a number of lattice conditions being weaker than in the Lie case. [ABSTRACT FROM AUTHOR]

The Schur -multiplier of Leibniz algebras is the Schur multiplier of Leibniz algebras defined relative to the Liezation functor. In this paper, we provide upperbounds for the dimensions of the c-nilpotent Schur -multiplier and c- stem cover of a finite-dimensional non-Lie Leibniz algebra. In the case of a -nilpotent non-Lie Leibniz algebra, we determine this upperbound in terms of the dimensions of its -center and -commutator. We also investigate the case of c-nilpotent Leibniz algebras of maximal class. [ABSTRACT FROM AUTHOR]