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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.

INTRODUCTION Lax Equation and IST Conserved Densities and Hamiltonian Structure Symmetry Aspects Observations LIE ALGEBRA Introduction Structure Constants and Basis of Lie Algebra Lie Groups and Lie Algebra Representation of a Lie Algebra Cartan-Killing Form Roots Space Decomposition Lie Groups: Finite and Infinite Dimensional Loop Groups Virasoro Group Quantum Tori Algebra Kac-Moody Algebra Serre's Approach to Kac-Moody Algebra Gradation Other Infinite Dimensional Lie Algebras PROLONGATION THEORY Introduction Sectioning of Forms The KdV Problem The Method of the Hall Structure Prolongation in (2+1) Dimension Method of Pseudopotentials Prolongation Structure and the Backlund Transformation Constant Coefficient Ideal Connections Morphisms and Prolongation Principal Prolongation Structure Prolongations and Isovectors Vessiot's Approach Observations CO-ADJOINT ORBITS Introduction The Kac-Moody Algebra Integrability Theorem: Adler, Kostant, Symes Superintegrable Systems Nonlinear Partial Differential Equation Extended AKS Theorem Space-Dependent Integrable Equation The Moment Map Moment Map in Relation to Integrable Nonlinear Equation Co-Adjoint Orbit of the Volterra Group SYMMETRIES OF INTEGRABLE SYSTEMS Introduction Lie Point and Lie Backlund Symmetry Lie Backlund Transformation Some New Ideas in Symmetry Analysis Non-Local Symmetries Observations HAMILTONIAN STRUCTURE Introduction Drinfeld Sokolob Approach The Lie Algebraic Approach Example of Hamiltonian Structure and Reduction Hamiltonian Reduction in (2+1) Dimension Hamiltonian Reduction of Drinfeld and Sokolov Kupershmidt's Approach Gelfand Dikii Formula Trace Identity and Hamiltonian Structure Symmetry and Hamiltonian Structure CLASSICAL r-MATRIX Introduction Double Lie Algebra Classical r-Matrix The Use of r-Matrix The r-Matrix and KP Equation

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.

We study the ground states and left-excited states of the A_{k-1} N=(2,0) little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory., Comment: 32 pages. Minor imprecisions and typos corrected. To appear in Nuclear Physics B

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.

We prove a determinant formula for a parabolic Verma module of a Lie superalgebra, previously conjectured by the second author. Our determinant formula generalizes the previous results of Jantzen for a parabolic Verma module of a (non-super) Lie algebra, and of Kac concerning a (non-parabolic) Verma module for a Lie superalgebra. The resulting formula is expected to have a variety of applications in the study of higher-dimensional supersymmetric conformal field theories. We also discuss irreducibility criteria for the Verma module., 24 pages

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.

Integrable systems with a linear periodic integral for the Lie algebra e(3) are considered. One investigates singularities of the Liouville foliation, bifurcation diagram of the momentum mapping, transformations of Liouville tori, topology of isoenergy surfaces and other topological properties of such systems.

We define Lie algebra cohomology associated with the half-Dirac operators for representations of rational Cherednik algebras and show that it has property described in the Casselman–Osborne Theorem by establishing a version of the Vogan's conjecture for the half-Dirac operators. Moreover, we study the relationship between Lie algebra cohomology and Dirac cohomology in analogy of the representations for semisimple Lie algebras.

The realification of the ( 2 n + 1 ) -dimensional complex Heisenberg Lie algebra is a ( 4 n + 2 ) -dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra sp ( n ) of derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the centre admits a compact Lie algebra of derivations. This also gives us the occasion to revisit a series of classic results, with the expressed aim of attracting the interest of a broader audience.

We give a new conceptual proof of the classification of cuspidal modules for the solenoidal Lie algebra. This classification was originally published by Y. Su in 2001. Our proof is based on the theory of modules for the solenoidal Lie algebras that admit a compatible action of the commutative algebra of functions on a torus.

The paper naturally continues series of works on identical relations of group rings, enveloping algebras, and other related algebraic structures. Let L be a Lie algebra over a field of characteristic p > 0 . Consider its symmetric algebra S ( L ) = ⊕ n = 0 ∞ U n / U n − 1 , which is isomorphic to a polynomial ring. It also has a structure of a Poisson algebra, where the Lie product is traditionally denoted by { , } . This bracket naturally induces the structure of a Poisson algebra on the ring s ( L ) = S ( L ) / ( x p | x ∈ L ) , which we call a truncated symmetric Poisson algebra. We study Lie identical relations of s ( L ) . Namely, we determine necessary and sufficient conditions for L under which s ( L ) is Lie nilpotent, strongly Lie nilpotent, solvable and strongly solvable, where we assume that p > 2 to specify the solvability. We compute the strong Lie nilpotency class of s ( L ) . Also, we prove that the Lie nilpotency class coincides with the strong Lie nilpotency class in case p > 3 . Shestakov proved that the symmetric algebra S ( L ) of an arbitrary Lie algebra L satisfies the identity { x , { y , z } } ≡ 0 if, and only if, L is abelian. We extend this result for the (strong) Lie nilpotency and the (strong) solvability of S ( L ) . We show that the solvability of s ( L ) and S ( L ) in case char K = 2 is different from other characteristics, namely, we construct examples of such algebras which are solvable but not strongly solvable. We use delta-sets for Lie algebras and the theory of identical relations of Poisson algebras. Also, we study filtrations in Poisson algebras and prove results on products of terms of the lower central series for Poisson algebras.

Given a finite-dimensional associative commutative algebra A over a field F, we define the structure of a Lie algebra using a nonzero derivation D of A. If A is a field and charF > 3; then the corresponding algebra is simple, presenting a nonisomorphic analog of the Zassenhaus algebra W 1(m).

In this paper, an explicit determinant formula is given for the Verma modules over the Lie algebra $W(2,2)$. We construct a natural realization of a certain vaccum module for the algebra $W(2,2)$ via the Weyl vertex algebra. We also describe several results including the irreducibility, characters and the descending filtrations of submodules for the Verma module over the algebra $W(2,2)$.

This is a very brief overview of some of the main ideas of algebra through history, from the decomposition of a binomial identity, to the degree of a complex number, Euler’s formula, transcendental numbers, skew fields, the Abel–Ruffini theorem, permutation groups, Lie algebras and more.

We prove that assosymmetric algebras under the Jordan product are Lie triple algebras. A Lie triple algebra is called special if it is isomorphic to a subalgebra of the plus-algebra of some assosymmetric algebra. We establish that the Glennie identity of degree 8 is valid for special Lie triple algebras, but not for all Lie triple algebras.

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.

We solve the M. V. Zaicev problem in the following sense: Any Noetherian semiprime special Lie algebra is embedded into algebra of matrices over commutative ring which is the direct sum of fields.

A nonassociative algebra is defined to be zeropotent if the square of any element is zero. Zeropotent algebras are exactly the same as anticommutative algebras when the characteristic of the ground field is not two. The class of zeropotent algebras properly contains that of Lie algebras. In this paper, we give a complete classification of three-dimensional zeropotent algebras over an algebraically closed field of characteristic not equal to two. By restricting the result to the subclass of Lie algebras, we can obtain a classification of three-dimensional complex Lie algebras, which is in accordance with the conventional one.

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.

The main objective of this study is five dimensional Lie algebra which describes the symmetry properties of a isothermal drift flux model of two phase flows. The classification provided is based on method given by Patera and Winternitz. Multi-dimensional optimal systems are obtained for the symmetry algebra of model. Further, using general theorem proved by Ibragimov we find several nonlocal conservation laws corresponding to every symmetry in one dimensional optimal system.

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.

Let L be a Lie algebra, and Der z (L) denote the set of all central derivations of L, that is, the set of all derivations of L mapping L into the center. In this paper, by using the notion of isoclinism, we study the center of Der z (L) for nilpotent Lie algebras with nilindex 2. We also give a characterization of stem Lie algebras by their central derivations. In fact we show that for non-abelian nilpotent Lie algebras of finite dimension and any nilpotent Lie algebra with nilindex 2 (not finite dimensional in general), Der z (L) is abelian if and only if L is a stem Lie algebra.

The algebras UTn(K)UTn(K) of the upper triangular matrices over a field K are of significant importance in the theory of algebras with polynomial identities. Group gradings on algebras appear in various areas and provide an indispensable tool in the study of the algebraic and combinatorial properties of the algebras in question. In this paper we consider the Lie algebra UTn(K)(−)UTn(K)(−) of all upper triangular matrices of order n . We study the group gradings on this algebra. It turns out that the gradings on the Lie algebra UTn(K)UTn(K) are much more intricate than those in the associative case. In this paper we describe the elementary gradings on the Lie algebra UTn(K)(−)UTn(K)(−). Finally we study the canonical grading on UTn(K)(−)UTn(K)(−) by the cyclic group ZnZn of order n. We produce a (finite) basis of the graded polynomial identities satisfied by this grading.

The loop and current Virasoro algebras are respectively the tensor product Lie algebras of the Virasoro algebra with the polynomial and Laurent polynomial algebra. In the present paper, the structures of dual Lie bialgebras of loop and current Virasoro algebras are investigated. As a result, a series of new infinite dimensional Lie algebras are obtained.

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.

A general method for constructing first-order symmetry operators for the stationary Schrodinger and Pauli equations is proposed. It is proven that the Lie algebra of these symmetry operators is a one-dimensional extension of some subalgebra of an e(3) algebra. We also assemble a classification of stationary electromagnetic fields for which the Schrodinger (or Pauli) equation admits a Lie algebra of first-order symmetry operators.

In this paper, we discuss some operators defined on Lie algebras for the purpose of deriving properties of some special functions. The method developed in this paper can also be used to study some other special functions of mathematical physics. We have established a general theorem concerning eigenvectors for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Further, using this result, we have obtained differential recurrence relations and differential equations for the extended Jacobi polynomials and the Gegenbauer polynomials. Results of many researchers; see for example Radulescu (1991), Mandal (1991), Pathan and Khan (2003), Humi, and the references therein, follow as special cases of our results.

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

In this paper, we mainly prove that each skew-symmetric biderivation of the Lie algebra is inner. As an application of biderivations, we will show that every linear commuting map on the Lie algebra has the form , where .

We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p ≥ 5 . For H 2 ( g ) = 0 , we explicitly describe the cocycles spanning H ⁎ 2 ( g ) , and in the case H 2 ( g ) ≠ 0 , we give a procedure to describe a basis for H ⁎ 2 ( g ) .

A supersymmetric integrable equation in (2+1) dimensions is constructed by means of the approach of the homogenous space of the super Lie algebra, where the super Lie algebra osp(3/2) is considered. For this (2+1) dimensional integrable equation, we also derive its Backlund transformation.

Cartan described some of the finite dimensional simple Lie algebras and three of the four series of simple infinite dimensional vectorial Lie algebras with polynomial coefficients as prolongs, which now bear his name. The rest of the simple Lie algebras of these two types (finite dimensional and vectorial) are, if the depth of their grading is greater than 1, results of generalized Cartan–Tanaka–Shchepochkina (CTS) prolongs. Here we are looking for new examples of simple finite dimensional modular Lie (super)algebras in characteristic 2 obtained as Cartan prolongs. We consider pairs (an (ortho-)orthogonal Lie (super)algebra or its derived algebra, its irreducible module) and compute the Cartan prolongs of such pairs. The derived algebras of these prolongs are simple Lie (super)algebras. We point out several amazing phenomena in characteristic 2: a supersymmetry of representations of certain Lie algebras, latent or hidden over complex numbers, becomes manifest; the adjoint representation of some simple Lie...

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 the classification of (n + 2)-dimensional metric n-Lie algebras in terms of some facts about n-Lie algebras.

In this paper, we first construct an analogue of the Sugawara operators for the twisted Heisenberg-Virasoro algebra. By using these operators, we show that every integrable highest weight module over an affine Lie algebra can be viewed as a unitary representation of the twisted Heisenberg-Virasoro algebra. As a by-product of our constructions, we give the unitary representations of the twisted Heisenberg-Virasoro algebra which have the central charges appearing in [1]. Our approach to obtain these central charges is different with that of [1].

Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x1, x2,..., xn) = \({\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type An, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.

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, a result for double index sequences is established by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space. This result is used to give new proofs of some properties of the Bessel and Tricomi functions of two indices. Further, this result is extended for multi-index sequences.

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.

For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded to the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfy the standard Lie identity of degree $5$. The problem of existence of such embedding is a long-standing problem (see \cite{YuPSt5,PoinsotAAM,Poinsot}), which is closely related to the Lie algebra of vector fields on the affine line (see \cite{YuPSt5}). It was solved by Razmyslov in \cite{YuPSt5} for simple Lie algebras satisfying this identity (see also \cite[Th. 16]{Poinsot}). We extend this result to prime (and semiprime) Lie algebras over a field of zero characteristic satisfying the standard Lie identity of degree~$5$. As an application, we prove that for any semiprime Lie algebra the standard identity $St_5$ implies all other identities of the Lie algebra of polynomial vector fields on the affine line. We also generalize some previous results about primeness of the Lie algebra of special derivations of a prime differential algebra to the case of non-unitary differential algebra.

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.