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We show that any semi-direct sum $L$ of Lie algebras with Levi factor $S$ must be perfect if the representation associated with it does not possess a copy of the trivial representation. As a consequence, all invariant functions of $L$ must be Casimir operators. When $S= \frak{sl}(2,\mathbb{K}),$ the number of invariants is given for all possible dimensions of $L$. Replacing the traditional method of solving the system of determining PDEs by the equivalent problem of solving a system of total differential equations, the invariants are found for all dimensions of the radical up to five. An analysis of the results obtained is made, and this lead to a theorem on invariants of Lie algebras depending only on the elements of certain subalgebras., 15 Pages

A G-grading on an algebra, where G is an abelian group, is called multiplicity-free if each homogeneous component of the grading is 1-dimensional. We introduce skew root systems of Lie type and skew root systems of Jordan type, and use them to construct multiplicity-free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp., Jordan) algebras are simple. Two families of skew root systems of Lie type (resp., of Jordan type) are constructed and the corresponding Lie (resp., Jordan) algebras are identified. This is a new approach to study abelian group gradings on Lie and Jordan algebras.

A Lie algebra L over a field $$\mathbb {F}$$ is said to be zero product determined (zpd) if every bilinear map with the property that $$f(x,y)=0$$ , whenever x and y commute, is a coboundary. The main goal of the paper is to determine whether or not some important Lie algebras are zpd. We show that the Galilei Lie algebra , where V is a simple $$\mathfrak {sl}_2$$ -module, is zpd if and only if $$\dim V =2$$ or $$\dim V$$ is odd. The class of zpd Lie algebras also includes the quantum torus Lie algebras $$\mathscr {L}_q$$ and $$\mathscr {L}^+_q$$ , the untwisted affine Lie algebras, the Heisenberg Lie algebras, and all Lie algebras of dimension at most 3, while the class of non-zpd Lie algebras includes the (4-dimensional) aging Lie algebra and all Lie algebras of dimension more than 3 in which only linearly dependent elements commute. We also give some evidence of the usefulness of the concept of zpd Lie algebra by using it in the study of commutativity preserving linear maps.

In this article, we present an explicit formula for the nilpotent multipliers of the direct sum of Lie algebras whose abelianisations are finite dimensional and, under some conditions, extend it for arbitrary Lie algebras. We also determine the nilpotent multipliers of some certain Lie algebras and obtain the exact structure of all c -capable nilpotent Lie algebras with derived subalgebra of dimension at most 1.

Let 𝒯 be a (n−1)-torsion free triangular algebra over a communicative ring ℛ and 𝒵(𝒯) be the center of 𝒯. Let GL:𝒯→𝒯 be a nonlinear generalized Lie n-derivation with associated Lie n-derivation L. ...

We establish the Gröbner–Shirshov bases theory for Lie differential [Formula: see text]-algebras. As an application, we give a linear basis of a free Lie differential Rota–Baxter algebra on a set.

This paper studies a generalized Benney–Luke equation with time-dependent coefficients using Lie symmetry methods. Lie group classification with respect to the time-dependent coefficients is performed by first deriving the equivalence group of transformations. We obtain the principal Lie algebra consisting of two translation symmetries and then using the classifying relations along with the equivalence transformations we obtain six distinct cases of the time-dependent coefficients for which the principal Lie algebra extends. Symmetry reductions and group-invariant solutions are obtained for all these six cases. Finally, conservation laws are derived for two cases of the time-dependent coefficients by employing the multiplier method.

In this paper, we investigate local Lie derivations of a certain class of operator algebras and show that, under certain conditions, every local Lie derivation of such an algebra is a Lie derivation.

In the paper, finite-dimensional real Lie algebras for which the centralizers of all nonzero element are Abelian are studied. These Lie algebras are also characterized by the transitivity condition for the commutation relation for two nonzero elements. A complete description of these Lie algebras up to isomorphism is given. Some results concerning the relationship between the aforementioned Lie algebras and the Lie algebras of vector fields whose orbits are one-dimensional are considered.

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent. In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank ≤ 4 with an idempotent are Jordan algebras. Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of e -stable power-associative algebras. We also prove that the algebra obtained by 2 -gametization process of a Lie triple algebra is a Lie triple one.

The aim of this work is to find some criteria for detecting the capability of a pair of Lie algebras. We characterize the exact structure of all pairs of capable Lie algebras in the class of abelian and Heisenberg ones. Among the other results, we also give some exact sequences on the Schur multiplier and exterior product of Lie algebras.

It is shown that the two-known series of rank one (Formula presented.) and rank two (Formula presented.) finite-dimensional solvable rigid Lie algebras with non-vanishing second cohomology can be extended to solvable rigid Lie algebras of arbitrary rank (Formula presented.) such that the cohomology is preserved exactly. For the second series, it is further proved that an extension decreasing the cohomology exists, hence leading to cohomologically rigid Lie algebras.

The classification of simple finite dimensional modular Lie algebras over algebraically closed fields of characteristic p > 3 (described by the generalized Kostrikin–Shafarevich conjecture) being completed due to Block, Wilson, Premet and Strade (with contributions from other researchers) the next major classification problems are those of simple finite dimensional modular Lie algebras over fields of characteristic 3 and 2. For the latter, the Kochetkov–Leites conjecture involved classification of Lie superalgebras and their inhomogeneous with respect to parity subalgebras, called Volichenko algebras. In characteristic 2, we consider the result of application of the functor forgetting the superstructure to the simple serial vectorial Lie algebras known to us and their Volichenko subalgebras.

In this paper we give an elementery construction of the Lie algebras of type E6(K) for fields K of characteristic two.

In this paper, we characterize the structure of the c-nilpotent multiplier of a pair of Lie algebras concerning the c-covering pairs and discuss some results on the existence of c-covers of a pair of Lie algebras. Moreover, we show that every nilpotent pair of Lie algebras of class at most k with non-trivial c-nilpotent multiplier does not admit any c-covering pair, if c>k. Also, we obtain the structure of c-covers of a c-capable pair of Lie algebras.

In this paper, we study a class of infinite-dimensional Lie algebras generalizing the constant r -term Krichever–Novikov type algebras. We associate vertex C ( ( z ) ) -algebras and their type zero modules to representations of these Lie algebras. An equivalence between the category of restricted modules for the Lie algebras and the category of type zero modules for the vertex C ( ( z ) ) -algebras is established.

Weyl was a universal mathematician whose fundamental contributions to mathematics encompassed all areas, and provided a unification seldom seen. His work on the theory of Lie groups was motivated by his life-long interest in quantum mechanics and relativity. WhenWeyl entered Lie theory, it mostly focussed on the infinitesimal, and he strove to bring in a global perspective. Time and again, Weyl’s ideas arising in one context have been adapted and applied to wholly new contexts. In 1925–26, Weyl wrote four epochal papers in representation theory of Lie groups which solved fundamental problems, and also gave birth to the subject of harmonic analysis of semisimple Lie groups. In these papers, Weyl proved complete reducibility theorems and introduced many techniques which have become the standard way to study representations of Lie groups and their various generalizations in the last seven decades. Weyl’s work covers several parts of mathematics, as well as parts of physics. In this article, we discuss mainly his contributions to the representation theory of Lie groups via the four papers mentioned above.

Lie algebras of order 3 constitute a generalization of Lie algebras and superalgebras. Throughout this paper the classification problem of filiform Lie algebras of order 3 is considered and therefore this work is a continuation papers seen in the literature. We approach this classification by extending Vergne’s result for filiform Lie algebras and by considering algebras of order 3 of high nilindex. We find the expression of the law to which any elementary filiform Lie algebra of order 3 is isomorphic.

The ℤ-grading determined by a long simple root of a rank n+1 affine Lie algebra over ℂ arises from a representation of a rank n semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation yields constructions that generalize the minuscule and adjoint algorithms as well as Kac’s construction of nontwisted affine Lie algebras.

We will characterize all finite dimensional Lie algebras with at most |F|2+|F|+2 centralizers, where F is the underlying field of Lie algebras under consideration.

We first adapt a method due to Skjelbred–Sund to classify p-nilpotent restricted Lie algebras. It turns out that any p-nilpotent restricted Lie algebra of dimension n can be constructed as a central extension of a p-nilpotent restricted Lie algebra of dimension n − 1 . We apply these techniques to classify all p-nilpotent restricted Lie algebras of dimension 5 over a perfect field of characteristic p ⩾ 5 .

The purpose of this paper is to derive some inequalities for dimension of the c-nilpotent multiplier of finite dimensional Lie algebras and their factor Lie algebras. We further obtain an inequality between dimensions of c-nilpotent multiplier of Lie algebra L and tensor product of a central ideal by its abelianized factor Lie algebra.

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A1, B, C, and BC.

In this paper, we present an explicit formula for the 2-nilpotent multiplier of a direct product of two Lie algebras.

In this paper, the authors study a class of generalized intersection matrix Lie algebras gim(Mn), and prove that its every finite-dimensional semisimple quotient is of type M(n, a, c, d). Particularly, any finite dimensional irreducible gim(Mn) module must be an irreducible module of Lie algebra of type M(n, a, c, d) and any finite dimensional irreducible module of Lie algebra of type M(n, a, c, d) must be an irreducible module of gim(Mn).

A flat Lorentzian Lie algebra is a left symmetric algebra endowed with a symmetric bilinear form of signature (−, +,…, +) such that left multiplications are skew-symmetric. In geometrical terms, a flat Lorentzian Lie algebra is the Lie algebra of a Lie group with a left-invariant Lorentzian metric with vanishing curvature. In this article, we show that any flat nonunimodular Lorentzian Lie algebras can be obtained as a double extension of flat Riemannian Lie algebras. As an application, we give all flat nonunimodular Lorentzian Lie algebras up to dimension 4.

The concept of n-Lie algebra were introduced by Filippov in 1987. One notes that not all properties of Lie algebras can be carried over to n-Lie algebras, as Williams showed in 2009. In the present article, among other results it is shown that the notions of isoclinism and isomorphism for two finite dimensional n-Lie algebras of the same dimension are equivalent. This was already done for ordinary Lie algebras by Moneyhun in 1994.

In terms of sandwich algebras, we obtain a classification of symmetrical simple Lie sheaves over an algebraically closed field of zero characteristic.

A pair of sequences of nilpotent Lie algebras denoted by Nn, 7 and Nn, 16 are introduced. Here, n denotes the dimension of the algebras that are defined for n ≥ 6; the first terms in the sequences are denoted by 6.7 and 6.16, respectively, in the standard list of six-dimensional Lie algebras. For each of them, all possible solvable extensions are constructed so that Nn, 7 and Nn, 16 serve as the nilradical of the corresponding solvable algebras. The construction continues Winternitz’ and colleagues’ program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.

In this paper we consider the universal central extension of a centerless Lie torus satisfying fgc condition and classify its irreducible integrable modules when the center acts non-trivially. These algebras (Lie tori) are naturally Zn-graded and we develop a one to one correspondence between their graded and non-graded modules. At the non-graded level their irreducible integrable modules turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism.

The purpose of the work is the classification of three-dimensional homogeneous spaces, allowing a normal connection, description of invariant affine connections on those spaces together with their curvature and torsion tensors, holonomy algebras. We consider only the case, when Lie group is solvable. The local classification of homogeneous spaces is equivalent to the description of the effective pairs of Lie algebras. We study the holonomy algebras of homogeneous spaces and find when the invariant connection is normal. Studies are based on the use of properties of the Lie algebras, Lie groups and homogeneous spaces and they mainly have local character.

In this paper, we present an algorithmic method to describe and classify filiform Lie algebras by considering the maximal dimension of abelian ideals and studying the relation between them and some known invariants of filiform Lie algebras. To do so, we have proved some new results about structure coefficients in the general law of filiform Lie algebras. Moreover, some mistakes are corrected in the already-known classification of filiform Lie algebras of dimension 8.

We show that untwisted respectively twisted conjugacy classes of a compact and simply connected Lie group which satisfy a certain integrality condition correspond naturally to irreducible highest weight representations of the corresponding affine Lie algebra. Along the way, we review the classification of twisted conjugacy classes of a simply connected compact Lie group G and give a description of their stabilizers in terms of the Dynkin diagram of the corresponding twisted affine Lie algebra

In this chapter, we discuss the representation theory of complex semisimple Lie algebras. While we give full details for \( \mathfrak{s}{{\mathfrak{l}}_{n}} \)(ℂ), we only sketch the details for other semisimple Lie algebras. For more details, we refer the reader to [26].

This chapter is on the structure theory of complex, semisimple Lie algebras. We give complete details for \( \mathfrak{s}{{\mathfrak{l}}_{n}} \)(ℂ) and give a brief account for other semisimple Lie algebras. For details, see [26].

In this paper we investigate the growth with respect to p of dimensions of irreducible representations of a semisimple Lie algebra g over Fp. More precisely, it is known that for p ≫ 0, the irreducibles with a regular rational central character λ and p-character χ are indexed by a certain canonical basis in the K0 of the Springer fiber of χ. This basis is independent of p. For a basis element, the dimension of the corresponding module is a polynomial in p. We show that the canonical basis is compatible with the two-sided cell filtration for a parabolic subgroup in the affine Weyl group defined by λ. We also explain how to read the degree of the dimension polynomial from a filtration component of the basis element. We use these results to establish conjectures of the second author and Ostrik on a classification of the finitedimensional irreducible representations of W-algebras, as well as a strengthening of a result by the first author with Anno and Mirkovic on real variations of stabilities for the derived category of the Springer resolution.

We prove a character formula for irreducible highest weight modules over a simple affine vertex algebra of level k, attached to a simple Lie algebra g, which are locally g-finite, in the cases when g is of type An andCn (n≥2) and k = −1. We also conjecture a character formula for types D4, E6, E7, E8 and levels k = −1, ..., −b, where b = 2, 3, 4, 6 respectively.

For any finite-dimensional Lie algebra we introduce the notion of Jordan–Kronecker invariants, study their properties, and discuss examples. These invariants naturally appear in the framework of the bi-Hamiltonian approach to integrable systems on Lie algebras and are closely related to Mischenko–Fomenko’s argument shift method. We also state a generalised argument shift conjecture and prove it for many series of Lie algebras.

The Lie algebras of order F have important applications for the fractional supersymmetry, and on the other hand the filiform Lie (super)algebras have very important properties into the Lie Theory. Thus, the aim of this work is to study filiform Lie algebras of order F which were introduced in Navarro (2014). In this work we obtain new families of filiform Lie algebras of order 3, in which the complexity of the problem rises considerably respecting to the cases considered in Navarro (2014).

Several improvements of the Kostrikin-Shafarevich method conjecturally producing all simple finite-dimensional Lie algebras over algebraically closed fields of any positive characteristic were rece ...

Let Γ be a generic subgroup of the multiplicative group C⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of ZN and an arbitrary finite order isometry of ZN preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As an application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras.

In this paper, we study the structure theory of a class of not-finitely graded Lie algebras related to generalized Heisenberg–Virasoro algebras. In particular, the derivation algebras, the automorphism groups and the second cohomology groups of these Lie algebras are determined.

International audience; In this study, we consider Lie algebras that admit para-Kählerand hyper-para-Kähler structures. We provide new charac-terizations of these Lie algebras and develop many methodsfor building large classes of examples. Previously, Bai consid-ered para-Kähler Lie algebras as left symmetric bialgebras.We reconsider this viewpoint and make improvements in or-der to obtain some new results. The study of para-Kähler andhyper-para-Kähler is intimately linked to the study of leftsymmetric algebras, particularly those that admit invariantsymplectic forms. In this study, we provide many new classesof left symmetric algebras and complete descriptions of all theassociative algebras that admit an invariant symplectic form.We also describe all four-dimensional hyper-para-Kähler Liealgebras.

The present paper contains two interrelated developments. First, the basic properties of the construction theory over the Steinberg Lie color algebras are developed in analogy with Steinberg Lie algebra case. This is done on the example of the central closed of the Steinberg Lie color algebras. The second development is that we define the first ɛ-cyclic homology group HC1(R, ɛ) of the Γ-graded associative algebra R (which could be seemed as the generalization of cyclic homology group and the ℤ/2ℤ-graded version of cyclic homology that was introduced by Kassel) to calculate the universal central extension of Steinberg Lie color algebras.

The complete classification of real solvable rigid Lie algebras possessing a nilradical of dimension at most six is given. Eleven new isomorphism classes of indecomposable algebras are obtained. It is further shown that the resulting solvable Lie algebras have a vanishing second Chevalley cohomology group, thus correspond to algebraically rigid Lie algebras.