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In this paper, the biderivations of Schrodinger–Virasoro Lie algebras are characterized. It is obtained a class of non-inner and non-skewsymmetric biderivations. As an application, the commutative post-Lie algebra structures on the Schrodinger–Virasoro Lie algebra are presented.

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

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.

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.

The loop Virasoro conformal superalgebra associated with the loop super-Virasoro algebra is constructed in the present paper. The conformal superderivation algebra of is completely determined, which is shown to consist of inner superderivations. And nontrivial free and free -graded -modules of rank two are classified. We also give a classification of irreducible free -modules of rank two and all irreducible submodules of each free -graded -module of rank two.

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 .

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.

The Grigorchuk and Gupta–Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [31] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [41] . Thus, we have examples of (self-similar) finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie algebras do not exist by a result of Martinez and Zelmanov [27] . The goal of the present paper is to construct analogues of the Grigorchuk and Gupta–Sidki groups in the world of Lie superalgebras of an arbitrary characteristic. We construct two examples R, Q of finitely generated self-similar Lie superalgebras over a field K of an arbitrary characteristic and study their properties and properties of their associative hulls. In case char K = 2 , these examples turn into restricted Lie algebras. The virtue of the present construction is that the Lie superalgebras have clear monomial bases. These Lie superalgebras have slow polynomial growth and are multigraded by multidegree in the generators. The Lie superalgebra R is Z 2 -graded, while Q has a multidegree Z 3 -gradation and a Z 2 -gradation. Both algebras R and Q have similar constructions, computations for R are simpler, but Q enjoys some more specific interesting properties. The Z 3 -components of Q lie inside an elliptic paraboloid in space, they are at most one-dimensional, thus, the Z 3 -grading of Q is fine. In the Z 2 -gradation of Q, all components Q n m , n , m ∈ Z , are infinite dimensional except for Q 00 = { 0 } . The Z 2 -gradation also yields a continuum of different decompositions into a direct sum of two locally nilpotent subalgebras Q = Q + ⊕ Q − . In both examples, ad a is nilpotent, a being even or odd with respect to the Z 2 -grading as Lie superalgebras. This property is an analogue of the periodicity of the Grigorchuk and Gupta–Sidki groups. In particular, Q is a nil finely-graded Lie superalgebra, which shows that an extension of a theorem due to Martinez and Zelmanov [27] for the Lie superalgebras of characteristic zero is not valid. Both Lie superalgebras are self-similar, contain infinitely many copies of themselves, let us also call them fractal due to this property.

The ground field in the text is of characteristic 2. The classification of modulo 2 gradings of simple Lie algebras is vital for the classification of simple finite-dimensional Lie superalgebras: with each grading, a simple Lie superalgebra is associated, see arXiv:1407.1695. No classification of gradings was known for any type of simple Lie algebras, bar restricted Jacobson-Witt algebras (i.e., the first derived of the Lie algebras of vector fields with truncated polynomials as coefficients) on not less than 3 indeterminates. Here we completely describe gradings modulo 2 for several series of Lie algebras and their simple relatives: of special linear series, its projectivizations, and projectivizations of the derived Lie algebras of two inequivalent orthogonal series (except for ${\mathfrak{o}}_��(8)$). The classification of gradings is new, but all of the corresponding superizations are known. For the simple derived Zassenhaus algebras of height $n>1$, there is an $(n-2)$-parametric family of modulo 2 gradings; all but one of the corresponding simple Lie superalgebras are new. Our classification also proves non-triviality of a deformation of a simple $3|2$-dimensional Lie superalgebra (new result).

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.

For a field F and a row-finite directed graph Γ, let L(Γ) be the associated Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra [L(Γ), L(Γ)] to be simple.

We introduce a new Lie algebra of infinite N × N matrices which have nonzero entries only in finite number of rows. We show that its subalgebra of matrices with trace 0 is uncountably dimensional simple Lie algebra.

Let S be a nonassociative k-algebra. By using the Lie triple system, we study the subspace I2(S) of the Steinberg Lie algebra st2(S) and give a necessary and sufficient condition for I2(S)=0.

We construct an action of the Lie superalgebras glm|nˆ(Cq) on the tensor product of the polynomial algebra and the exterior algebra with infinitely many variables. This construction is involving one parameter μ and we show that it is irreducible if and only if μ is nonzero. We also discuss the module structure and irreducibility over a Lie subsuperalgebra graded by the root system A(m−1,n−1).

An element $x$ of a Lie algebra $L$ over the field $F$ is extremal if $[x,[x,L]]=Fx$. Under minor assumptions, it is known that, for a simple Lie algebra $L$, the extremal geometry ${\cal{E}}(L)$ is a subspace of the projective geometry of $L$ and either has no lines or is the root shadow space of an irreducible spherical building $\Delta$. We prove that if $\Delta$ is of simply-laced type, then $L$ is a quotient of a Chevalley algebra of the same type., Comment: 24 pages

We obtain the superconformal algebra associated to a sigma model with target a manifold with G 2 holonomy, i.e., the Shatashvili-Vafa G 2 algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra D(2, 1; α) for α = 1. We produce the complete family of W-algebras SW (3/2, 3/2, 2) (extensions of the N = 1 superconformal algebra by two primary supercurrents of conformal weight 3/2 and 2 respectively) as a quantum Hamiltonian reduction of D(2, 1; α). As a corollary we find a free field realization of the Shatashvili-Vafa G 2 algebra, and an explicit description of the screening operators.

This paper is concerned with the Lie superalgebra S(n, m) of special type over a field of prime characteristic. We construct the modular Lie superalgebra S(n, m) and discuss some properties of this algebra. Then the derivation superalgebra of S(n, m) is determined.

Let L be a Lie algebra of Block type over ℂ with basis {Lα,i | α,i ∈ ℤ} and brackets [Lα,i,Lβ,j]=(β(i+1)-α(j+1)) Lα+β,i+j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with ℂ[∂]-basis {Lα(w) | α ∈ ℤ} and λ-brackets [Lα(w)λ Lβ(w)]= (α∂+(α+β)λ) Lα+β(w). Finally, we give a classification of free intermediate series B-modules.

The variety generated by a three-dimensional simple Lie algebra over a field of characteristic zero was studied rather well. The study of this variety is continued in the paper and a formula for calculation of its colengths is presented.

Certain important results concerning p-envelopes of modular Lie algebras are generalized to the super-case. In particular, any p-envelope of the Lie algebra of a Lie superalgebra can be naturally extended to a restricted envelope of the Lie superalgebra. As an application, a theorem on the representations of Lie superalgebras is given, which is a super-version of Iwasawa's theorem in Lie algebra case. As an example, the minimal restricted envelopes are computed for three series of modular Lie superalgebras of Cartan type.

Let K be a field of characteristic 0 and let W 1 be the Lie algebra of the derivations of the polynomial ring K [ t ] . The algebra W 1 admits a natural Z -grading. We describe the graded identities of W 1 for this grading. It turns out that all these Z -graded identities are consequences of a collection of polynomials of degree 1, 2 and 3 and that they do not admit a finite basis. Recall that the “ordinary” (non-graded) identities of W 1 coincide with the identities of the Lie algebra of the vector fields on the line and it is a long-standing open problem to find a basis for these identities. We hope that our paper might be a step to solving this problem.

The post-Lie algebra is an enriched structure of the Lie algebra introduced by Vallette. In this paper we give a complete classification of post-Lie algebra structures on solvable Lie algebra t ( 2 , C ) , the Lie algebra of 2 × 2 upper triangular matrices. Furthermore, we discuss their isomorphism classes and obtain one necessary and sufficient condition.

We classify deformations of the standard embedding of the Lie algebra s l ( 2 ) into both the Lie algebra Ψ D O L of pseudodifferential operators with polynomial coefficients and the Poisson Lie algebra P , we prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem for the case of the standard embedding of the orthosymplectic Lie superalgebra o s p ( n | 2 ) on the ( 1 , n ) -dimensional superspace R 1 | n into the Lie superalgebra S Ψ D O ( n ) of superpseudodifferential operators with polynomial coefficients, where n = 1 , 2 getting the necessary and sufficient conditions for its integrability. Finally, by using the contract procedure we deduce similar results for the standard embedding into the Poisson Lie superalgebra S P ( n ) .

Let M(A) be a free metabelian Lie algebra with a finite generating set A over an algebraically closed field F of characteristic zero, in which the problem of there being solutions to a system of linear equations is decided algorithmically, and let M′(A) be the derived subalgebra of M(A). We present an algorithm for finding width of elements in M′(A).

In this paper, we study an infinite-dimensional Lie algebra ℬq, called the q-analog Klein bottle Lie algebra. We show that ℬq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of ℬq are also determined.

We prove the local solvability of the prime radical of a Lie algebra with descending chain condition for the ideals. We note that an analog of this statement holds also for the graded Ω-groups with finiteness condition.

Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra.

Symmetries of the Levy-Leblond equation are investigated beyond the standard Lie framework. It is shown that the equation has two remarkable symmetries. One is given by the super Schrodinger algebra and the other by a Z2×Z2 graded Lie algebra. The Z2×Z2 graded Lie algebra is achieved by transforming bosonic into fermionic operators in the super Schrodinger algebra and introducing second order differential operators as generators of symmetry.

We investigate the conditions under which the smash product of an (ordinary or restricted) enveloping algebra and a group algebra is Lie solvable or Lie nilpotent.

Let be the Lie algebra of all n × n strictly block upper triangular matrices over a field . In this paper, we explicitly describe all Lie triple derivations of when char()≠2.

The purpose of the article is to describe all 2-dimensional subalgebras and all corresponding reductive pairs of Lie algebra g of all 2 × 2 real matrices. This Lie algebra is 4-dimensional as a vector space, it’s not simple, and it’s not solvable. The evaluation procedure utilizes canonical bases for subspaces that were introduced. Part I of the article contains necessary basic information. In Part II, all 2-dimensional subalgebras of the given Lie algebra g are classified. All reductive pairs { , m} with 2-dimensional subalgebras h are found in Part III. The separate article contributes classification of all 3-dimensional subalgebras and its reductive pairs. Together, both articles give the total classification of all subalgebras and all reductive pairs of Lie algebra g.

Lie algebras and quantum groups are not usually studied by an undergraduate. However, in the study of these structures, there are interesting questions that are easily accessible to an upper-level undergraduate. Here we look at the expansion of a nested set of brackets that appears in relations presented in a paper of Lum on toroidal algebras. We illuminate certain terms that must be in the expansion, providing a partial answer for the closed form. Lie algebras and quantum groups are not topics that you are apt to hear undergraduates math majors discussing in their spare time. However, there are a surprising number of nontrivial questions in this area that are undergraduate appropriate. In this paper, we will give a brief overview of the broad mathematical setting, and then discuss an accessible problem that involves expanding a nested set of brackets. Lie algebras, their universal enveloping algebras and quantum groups are a fundamental part of representation theory that have many applications within mathematics and mathematical physics. Lie algebras and Lie groups were originally discovered by Sophus Lie in the late nineteenth century [Borel 2001]. Given a Lie algebra, we associate a unique associative algebra called the universal enveloping algebra. In 1985, Jimbo and Drinfeld discovered q-analogues of these universal enveloping algebras called “quantum groups”, which have been a recent area of study (see [Lusztig 1993]). In order to find the quantum analogue of a Lie algebra it is often desirable to understand the defining relationships of the Lie algebra inside of its universal enveloping algebra. The motivation for this project came from a paper by Lum in which he gives a nice presentation of a toroidal Lie algebra that could be useful in understanding this Lie algebra’s quantum group [Lum 1998]. All of these relations utilize a nested set of brackets called t.k/. For simplicity, we have modified t.k/ by a scalar. In this paper we seek to understand the expansion of this object.

Consider g a Lie algebra graded by an arbitrary set I (set grading). We show that g decomposes as the sum of well-described graded ideals plus (maybe) a certain linear subspace. Under mild conditions, the simplicity of g is characterized and it is shown that the above decomposition is actually the direct sum of the family of its minimal graded ideals (each one being a simple set-graded Lie algebra).

We consider the Lie algebra $\mathfrak{g} = \mathfrak{p}_n $ of (n + 1) × (n + 1) matrices with zeros in the last row. This algebra has received the name of mirabolic; it has many remarkable properties and plays an important role in representation theory. In this paper we study open coadjoint orbits for the corresponding Lie group P n .

We show that each irreducible, transitive finite-dimensional graded Lie algebra over a field of prime characteristic p contains an initial subalgebra in which the pth power of the adjoint transformation associated with any element in the lowest gradation space is zero.

In the present article we shall investigate the Lie bialgebra structures on the Lie algebra , which are shown to be triangular coboundary.

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.

Let L be a restricted Lie superalgebra with its restricted enveloping algebra u ( L ) over a field F of characteristic p > 2 . Then u ( L ) can be viewed both as a Lie algebra and a Lie superalgebra. We characterize L when u ( L ) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.

In this article, by using Composition-Diamond lemma for Lie algebras, we give a Grobner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application, we obtain a normal form for such a Lie algebra.

Let G be a group. The Lie algebra sl 2 of 2 × 2 traceless matrices over a field K can be endowed up to isomorphism, with three distinct non-trivial G-gradings induced by the groups Z 2 , Z 2 × Z 2 and Z . It has been recently shown (Koshlukov, 2008 [8] ) that for each grading the ideal of G-graded identities has a finite basis. In this paper we prove that when char ( K ) = 0 , the algebra sl 2 endowed with each of the above three gradings has an ideal of graded identities Id G ( sl 2 ) satisfying the Specht property, i.e., every ideal of graded identities containing Id G ( sl 2 ) is finitely based.

The main aim of this paper is to give a description of Lie superhomomorphisms from a Lie ideal of the skew elements of a superalgebra with superinvolution into a unital superalgebra. As a consequence we improve a well-known result on Lie homomorphisms of rings obtained by Beidar et al. in 2001.

Let L be a Lie superalgebra with its enveloping algebra U(L) over a field F. A polynomial identity is called non-matrix if it is not satisfied by the algebra of 2×2 matrices over F. We characterize L when U(L) satisfies a non-matrix polynomial identity. We also characterize L when U(L) is Lie solvable, Lie nilpotent, or Lie super-nilpotent.

The famous Poincare-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

It is well known that the validity of the so called Lenard–Magri scheme of integrability of a bi-Hamiltonian PDE can be established if one has some precise information on the corresponding 1st variational Poisson cohomology for one of the two Hamiltonian operators. In the first part of the paper we explain how to introduce various cohomology complexes, including Lie superalgebra and Poisson cohomology complexes, and basic and reduced Lie conformal algebra and Poisson vertex algebra cohomology complexes, by making use of the corresponding universal Lie superalgebra or Lie conformal superalgebra. The most relevant are certain subcomplexes of the basic and reduced Poisson vertex algebra cohomology complexes, which we identify (non-canonically) with the generalized de Rham complex and the generalized variational complex. In the second part of the paper we compute the cohomology of the generalized de Rham complex, and, via a detailed study of the long exact sequence, we compute the cohomology of the generalized variational complex for any quasiconstant coefficient Hamiltonian operator with invertible leading coefficient. For the latter we use some differential linear algebra developed in the Appendix., National Science Foundation (U.S.), ERC (Advanced Grant)