Bai Ruipu, Zhang yanyan
In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras, we also give the relations between elementary n-algebras and E-algebras.
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By definition, a quadratic Lie superalgebra is a Lie superalgebra endowed with a non-degenerate supersymmetric bilinear form which satisfies the even and invariant properties. In this paper we calculate all of the second cohomology group of elementary quadratic Lie superalgebras which have been classified in \cite{DU14} by applying the super-Poisson bracket on the super exterior algebra. Besides, we give the classification of 8-dimensional solvable quadratic Lie superalgebras having 6-dimensional indecomposable even part. The method is based on the double extension and classification results o…
We initiate the investigation of the projective variety $E(r,g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $g$ for some $r > 0$ and demonstrate that this variety encodes considerable information about the representations of $g$. For various choices of $g$ and $r$, we identify the geometric structure of $E(r,g)$. We show that special classes of (restricted) representations of $g$ lead to algebraic vector bundles on $E(r,g)$. For $g = Lie(G)$ the Lie algebra of an algebraic group $G$, rational representations of $G$ enable us to realize familiar algebraic vecto…
The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any perfect field.
A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $Φ$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $ϕ$-free algebras respectively, then it is …
We give a constructive elementary proof for the fact that any K-automorphism of the full nxn matrix algebra over a field K is conjugation by some invertible nxn matrix A over K.
We find a basis for the $G$-graded identities of the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of characteristic $p>0$ with an elementary grading such that the neutral component corresponds to the diagonal of $M_n(K)$.
Embedding a graded matrix algebra into an elementary graded
本文证明任意全矩阵环的子代数（基于任意分次除环）可嵌入到初等分次矩阵环中，从而统一处理非初等分次结构。
A short construction of the Lie algebra $G_2(K)$ over fields $K$ of characteristic $2$
本文提供了一种在特征为2的域上构造G₂(K)型李代数的简短基础方法。
8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char $\ne 2, 3$
本文在特征不等于2,3的代数闭域上，利用代数几何方法和基础线性代数，对8维2步 nil 拉格朗日代数进行了自包含的分类。
In this note we prove that elementary maps on triangular algebras are automically additive.
Graded polynomial identities for the Lie algebra of upper triangular matrices of order 3
本文计算了3阶上三角矩阵李代数在初等分次下的分次多项式恒等式及其分次余维数序列。
Let $E$ be the Grassmann algebra of an infinite dimensional vector space $L$ over a field of characteristic zero. In this paper, we study the $\mathbb{Z}$-gradings on $E$ having the form $E=E_{(r_{1},r_{2}, r_{3})}^{(v_{1},v_{2}, v_{3})}$, in which each element of a basis of $L$ has $\mathbb{Z}$-degree $r_{1}$, $r_{2}$, or $r_{3}$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connect…
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.
Wedderburn's theorem on the structure of finite dimensional semisimple algebras is proved by using minimal prerequisites.
Universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie algebras are constructed. It is proved that the pairs of varieties (Lie Rota-Baxter algebras of zero weight,preLie algebras) and (Lie Rota-Baxter algebras of nonzero weight,post-Lie algebras) are PBW-pairs and the variety of Lie Rota-Baxter algebras is not Schreier.
Orthogonal Lie algebras in dimension 4 are identified as current Lie algebras, thus producing a natural decomposition for them over any field.
We study post-Lie algebra structures on pairs of Lie algebras $(\mathfrak{g},\mathfrak{n})$, motivated by nil-affine actions of Lie groups. We prove existence results for such structures depending on the interplay of the algebraic structures of $\mathfrak{g}$ and $\mathfrak{n}$. We consider the classes of simple, semisimple, reductive, perfect, solvable, nilpotent, abelian and unimodular Lie algebras. Furthermore we consider commutative post-Lie algebra structures on perfect Lie algebras. Using Lie algebra cohomology we prove that such structures are trivial in several cases. We classify commu…
We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on parabolic subalgebras of simple Lie algebras.
We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify filiform Lie algebras admitting a non-singular prederivation but no non-singular derivation. We prove that any 4-step nilpotent Lie algebra admits a non-singular prederivation. The results can be applied to construct affine structures on Lie algebras.
We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.
A twisted generalization of Lie-Yamaguti algebras, called Hom-Lie-Yamaguti algebras, is defined. Hom-Lie-Yamaguti algebras generalize Hom-Lie triple systems (and susequently ternary Hom-Nambu algebras) and Hom-Lie algebras in the same way as Lie-Yamaguti algebras generalize Lie triple systems and Lie algebras. It is shown that the category of Hom-Lie-Yamaguti algebras is closed under twisting by self-morphisms. Constructions of Hom-Lie-Yamaguti algebras from classical Lie-Yamaguti algebras and Malcev algebras are given. It is observed that, when the ternary operation of a Hom-Lie-Yamaguti alge…
In this paper, we characterize the biderivations of Schrödinger-Virasoro Lie algebra. We get a classes of non-inner and non-skewsymmetric biderivations. As application, we characterize the commutative post-Lie algebra structures on Schrödinger-Virasoro Lie algebra.
We propose a notion of a super n-Lie algebra and construct a super n-Lie algebra with the help of a given binary super Lie algebra which is equipped with an analog of a supertrace. We apply this approach to the super Lie algebra of a Clifford algebra with even number of generators and making use of a matrix representation of this super Lie algebra given by a supermodule of spinors we construct a series of super 3-Lie algebras labeled by positive even integers.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
M. Boucetta introduced the notion of pseudo-Riemannian Lie algebra in [2] when he studied the line Poisson structure on the dual of a Lie algebra. In this paper, we redefine pseudo-Riemannian Lie algebra, which, in essence, is a class of new Lie admissible algebras and prove that all pseudo-Riemannian Lie algebras are solvable. Using our main result and method, we prove some of M. Boucetta's results in [2, 3] in a simple and new way, then we give an explicit construction of Riemann-Lie algebras and a classification of pseudo-Riemannian Lie algebras of dimension 2 and 3.
We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.
The aim of this paper is to introduce $n$-ary BiHom-algebras, generalizing BiHom-algebras. We introduce an alternative concept of BiHom-Lie algebra called BiHom-Lie-Leibniz algebra and study various type of $n$-ary BiHom-Lie algebras and BiHom-associative algebras. We show that $n$-ary BiHom-Lie-Leibniz algebra can be represented by BiHom-Lie-Leibniz algebra through fundamental objets. Moreover, we provide some key constructions and study $n$-ary BiHom-Lie algebras induced by $(n-1)$-ary BiHom-Lie algebras.
We study classical R-matrices D for Lie algebras L such that D is also a derivation of L. This yields derivation double Lie algebras (L,D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.
Oscillator Lie algebras are the only non commutative solvable Lie algebras which carry a bi-invariant Lorentzian metric. In this paper, we determine all the Poisson structures, and in particular, all symmetric Leibniz algebra structures whose underlying Lie algebra is an oscillator Lie algebra. We give also all the symmetric Leibniz bialgebra structures whose underlying Lie bialgebra structure is a Lie bialgebra structure on an oscillator Lie algebra. We derive some geometric consequences on oscillator Lie groups.
We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore we show that all commutative post-Lie algebra structures on affine Kac--Moody Lie algebras are "almost trivial".
For $a,b\in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skewsymmetric) non-inner biderivations. As an applications, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.
We investigate a new class of Lie algebras, which are tame locally extended affine Lie algebras of nullity 1. It is an infinite-rank analog of affine Lie algebras, and their centerless cores are a local version of loop algebras. Such algebras are called locally affine Lie algebras and locally loop algebras. We classify both of them.
In this paper, we generalize the Lyndon-Shirshov words to Lyndon-Shirshov $Ω$-words on a set $X$ and prove that the set of all non-associative Lyndon-Shirshov $Ω$-words forms a linear basis of the free Lie $Ω$-algebra on the set $X$. From this, we establish Gröbner-Shirshov bases theory for Lie $Ω$-algebras. As applications, we give Gröbner-Shirshov bases for free $λ$-Rota-Baxter Lie algebras, free modified $λ$-Rota-Baxter Lie algebras and free Nijenhuis Lie algebras and then linear bases of such three free algebras are obtained.
Post-Lie conformal algebra structures on Lie conformal algebras
本文研究后李拟代数结构，建立其与李拟代数上权重为1的罗塔-巴克斯算子的等价关系，并分类了B(q)和W(b)两类李拟代数上的后李拟代数结构。
The aim of this paper is to introduce the notion of intuitionistic fuzzy Lie subalgebras and intutionistic fuzzy Lie ideals of n-Lie algebras. It is a generalization of intuitionistic fuzzy Lie algebras. Then, we investigate some of characteristics of intutionistic fuzzy Lie ideals (resp. subalgebras) of n-Lie algeras. Finally, we define the image and the inverse image of intuitionistic fuzzy Lie subalgebra under n-Lie algebra homomorphism. The properties of intuitionistic fuzzy n-Lie subalgebras and intuitionistic fuzzy Lie ideals under homomorphisms of n-Lie algebras are studied. Finally, we…
On quadratic Lie algebras containing the Heisenberg Lie algebra
本文研究了包含海森堡李代数的二次李代数的构造方法及性质，探讨了其扩展条件和不变度量的存在性。
An elementary proof of the Artin--Springer theorem for generalized quadratic forms over quaternion algebras
本文为四元数除代数上各向异性广义二次型在奇数次域扩张下保持各向异性提供了初等证明。
We give an efficient solution to the following problem: Given $X_1, \ldots X_d$ and $Y$ some $n$ by $n$ matrices can we determine if $Y$ is in the unital algebra generated by $X_1, \ldots, X_d$ as a subalgebra of all $n$ by $n$ matrices? The solution also gives an easy method for computing the dimension of this algebra.
What are all rings $R$ for which $R^*$ (the group of invertible elements of $R$ under multiplication) is an elementary abelian $p$-group? We answer this question for finite-dimensional commutative $k$-algebras, finite commutative rings, modular group algebras, and path algebras. Two interesting byproducts of this work are a characterization of Mersenne primes and a connection to Dedekind's problem.
We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the Lie algebra gl_N(C). In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova (math.RT/0610049).
This paper presents a Coq formalization of linear algebra over elementary divisor rings, that is, rings where every matrix is equivalent to a matrix in Smith normal form. The main results are the formalization that these rings support essential operations of linear algebra, the classification theorem of finitely presented modules over such rings and the uniqueness of the Smith normal form up to multiplication by units. We present formally verified algorithms computing this normal form on a variety of coefficient structures including Euclidean domains and constructive principal ideal domains. W…
By elementary and direct calculations the vanishing of the (algebraic) second Lie algebra cohomology of the Witt and the Virasoro algebra with values in the adjoint module is shown. This yields infinitesimal and formal rigidity or these algebras. The first (and up to now only) proof of this important result was given 1989 by Fialowski in an unpublished note. It is based on cumbersome calculations. Compared to the original proof the presented one is quite elegant and considerably simpler.
We construct the Lie algebra of an n-Lie algebra and we also define the notion of cohomology of an n-Lie algebra.
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced $3$-Lie algebras.
The notions of Lie color 2-algebras and 2-term color L-infty-algebras over a group-graded vector space are introduced and studied. It is proved that the category of Lie color 2-algebras and the category of 2-term color L1-algebras are equivalent. We construct Lie color 2-algebras from omni-Lie color algebras and Leibniz color algebras. Some example of $Z^2_2$ -graded Lie color 2-algebras are given.
Representations of color Hom-Lie algebras are reviewed, and it is shown that there exist a series of coboundary operators. We also introduce the notion of a color omni-Hom-Lie algebra associated to a vector space and an even invertible linear map. We show how regular color Hom-Lie algebra structures on a vector space can be characterized. Moreover, it is shown that the underlying algebraic structure of the color omni-Hom-Lie algebra is a color Hom-Leibniz algebra.
We study post-Lie algebra structures on pairs of Lie algebras (g,n), and prove existence results for the case that one of the Lie algebras is semisimple. For semisimple g and solvable n we show that there exist no post-Lie algebra structures on (g,n). For semisimple n and certain solvable g we construct canonical post-Lie algebra structures. On the other hand we prove that there are no post-Lie algebra structures for semisimple n and solvable, unimodular g. We also determine the generalized $(\al,\be,\ga)$-derivations of n in the semisimple case. As an application we classify post-Lie algebra …
We introduce post-Lie algebra structures on pairs of Lie algebras $(\Lg,\Ln)$ defined on a fixed vector space $V$. Special cases are LR-structures and pre-Lie algebra structures on Lie algebras. We show that post-Lie algebra structures naturally arise in the study of NIL-affine actions on nilpotent Lie groups. We obtain several results on the existence of post-Lie algebra structures, in terms of the algebraic structure of the two Lie algebras $\Lg$ and $\Ln$. One result is, for example, that if there exists a post-Lie algebra structure on $(\Lg,\Ln)$, where $\Lg$ is nilpotent, then $\Ln$ must …
The purpose of this paper is to introduce twisted $\mathcal{O}$-operators on $3$-Lie algebras. We define a cohomology of a twisted $\mathcal{O}$-operator $T$ as the Chevalley-Eilenberg cohomology of a certain $3$-Lie algebra induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted $\mathcal{O}$-operators from cohomological points of view. Furthermore, we introduce and study $3$-NS-Lie-algebras as the underlying structure of twisted $\mathcal{O}$-operators on $3$-Lie algebras. Finally, we investigate twisted $\mathcal{O}$-o…
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
Let $Σ$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(Σ)$, $\mathcal{U}(Σ)$ and $\mathcal{L}(Σ)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(Σ)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(Σ)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(Σ)$ is invariant under seed mutations when each exchange polynomial…
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. Reyn…
This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define for each Lie algebra a cohomology with values in a Lie-admissible module. This permits to study some deformations of Lie algebras in the category of Lie-admissible algebras. Lastly we study the tensor product between these operads and their dual operads. As application we constr…
In this paper, we define a class of 3-algebras which are called 3-Lie-Rinehart algebras. A 3-Lie-Rinehart algebra is a triple $(L, A, ρ)$, where $A$ is a commutative associative algebra, $L$ is an $A$-module, $(A, ρ)$ is a 3-Lie algebra $L$-module and $ρ(L, L)\subseteq Der(A)$. We discuss the basic structures, actions and crossed modules of 3-Lie-Rinehart algebras and construct 3-Lie-Rinehart algebras from given algebras, we also study the derivations from 3-Lie-Rinehart algebras to 3-Lie $A$-algebras. From the study, we see that there is much difference between 3-Lie algebras and 3-Lie-Rineha…
In this paper, first we introduce the notions of relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras and $3$-post-Lie algebras. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weight induces a $3$-post-Lie algebra naturally. Conversely, a $3$-post-Lie algebra gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Then we construct an $L_\infty$-algebra whose Maur…
We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by the same data: a metric Lie algebra with a skew-symmetric derivation. This is the same data defining a one-dimensional double extension of the metric Lie algebra and, indeed, bargmannian Lie algebras coincide with such double extensions, containing carrollian Lie algebras as an ideal and projecting to galilean Lie algebras. This sets up a canonical correspond…
Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that Lie triple system universal central extension is isomorphic either to Lie or Leibniz algebra universal central extension considered as a Lie triple system in appropriate way.
In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of t…
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