Alberto Elduque
Some simple Lie superalgebras, specific of characteristic 3, defined by S. Bouarroudj and D. Leites, are shown to be related to the simple alternative and commutative superalgebras discovered by I.P. Shestakov.
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The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group of degree 4 on this construction are given. By means of these actions, the models provided by the Tits construction are related to models of the exceptional Lie algebras obtained from two different types of structurable algebras. Some models of exceptional Lie superalgebras are discussed too.
The classical Tits construction of the exceptional simple Lie algebras has been extended in a couple of directions by using either Jordan superalgebras or composition superalgebras. These extensions are reviewed here. The outcome has been the discovery of some new simple modular Lie superalgebras.
The derivations of the Cheng-Kac Jordan superalgebras are studied. It is shown that, assuming -1 is a square in the ground field, the Lie superalgebra of derivations of a Cheng-Kac Jordan superalgebra is isomorphic to the Lie superalgebra obtained from a simpler Jordan superalgebra (a Kantor double superalgebra of vector type) by means of the Tits-Kantor-Koecher construction. This is done by exploiting the S4-symmetry of the Cheng-Kac Jordan superalgebra.
Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan superalgebras.
Two new simple modular Lie superalgebras are obtained in characteristics 3 and 5, which share the property that their even parts are orthogonal Lie algebras and the odd parts their spin modules. The characteristic 5 case is shown to be related, by means of a construction of Tits, to the exceptional ten dimensional Jordan superalgebra of Kac.
Cyclic homology of Jordan superalgebras and related Lie superalgebras
本文研究Jordan超代数的循环同调与其Tits-Kantor-Koecher Lie超代数二阶上同调的关系，重点处理Kantor二重情形，并将结果用于任意系数环上Hamilton型与接触型Lie超代数的二阶上同调和泛中心扩张计算。
This paper considers the multiplicative Hom-Lie superalgebra structures on infinite dimensional simple Lie superalgebras of vector fields with characteristic zero. The main result is that there is only the multiplicative Hom-Lie superalgebra structure on these Lie superalgebras.
The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of $(α^s,β^r)$-derivation, $(α^s,β^r)$-quasiderivation and generalized $(α^s,β^r)$-derivation of 3-BiHom-Lie superalgebras, and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of BiHom $3$-Lie superalgebras. We also explore the construction of $3$-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras.
(Multiplicative) Hom-Lie-Yamaguti superalgebras which generalize Hom-Lie supertriple systems (and subsequently ternary multiplicative Hom-Nambu superalgebras) and Hom-Lie superalgebras in the same way as Lie-Yamaguti superalgebras [Frac] generalize Lie supertriple systems and Lie superalgebras are defined. We show that the category of (multiplicative) Hom-Lie-Yamaguti superalgebras is closed under twisting by self-morphisms. Construction of some examples of Hom-Lie-Yamaguti superalgebras is given. The notion of an $n^{th}-$derived (binary) Hom-superalgebras is extended to the one of an $n^{th}…
In this paper, enveloping Lie superalgebras of Bol superalgebras are introduced. The notion of Killing-Ricci forms and invariant forms of these superalgebras are investigated as generalization of the ones of Bol algebras.
Throughout this paper, we will study Rota-Baxter operators and super O-operator of associative BiHom superalgebras, BiHom Lie superalgebras, BiHom pre-Lie superalgebras and BiHom-L-dendriform superalgebras. Then we give some properties of BiHom pre-Lie superalgebras constructed from associative BiHom superalgebras, BiHom Lie superalgebras and BiHom-L-dendriform superalgebras. As applications, we study the theory of cohomology groups of BiHom-pre-Lie superalgebras.
The purpose of this paper is to study the relationships between a Hom-Lie superalgebra and its induced 3-ary-Hom-Lie superalgebra. We provide an overview of the theory and explore the structure properties such as ideals, center, derived series, solvability, nilpotency, central extensions, and the cohomology.
The aim of this paper is to introduce the notion of restricted Hom- Lie superalgebras. This class of algebras is a generalization of both restricted Hom-Lie algebras and restricted Lie superalgebras. In this paper, we present a way to obtain restricted Hom-Lie superalgebras from the classical restricted Lie superalgebras along with algebra en- domorphisms. Homomorphisms relations between restricted Hom-Lie superalgebras are defined and studied. Also, we obtain some proper- ties of p-maps and restrictable Hom-Lie superalgebras.
The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra $\langle A, Δ\rangle$, it is possible to construct a Lie coalgebra $\langle L(A), Δ_{L}\rangle$. Moreover, any dual algebra of the coalgebra $\langle L(A), Δ_{L}\rangle$ corresponds to a Lie algebra that can be determined from the dual algebra for $\langle A,Δ\rangle$, following the Kantor--Koecher--Tits process. The structure of subcoalgebras and coideals of the coalgebra $\langle L(A), Δ_{L}\rangle$ is characterized.
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…
The aim of this paper is to introduce the notion of BiHom-Lie superalgebras. This class of algebras is a generalization of both BiHom-Lie algebras and Hom-Lie superalgebras. In this article, we first present two ways to construct BiHom-Lie superalgebras from BiHom-associative superalgebras and Hom-Lie superalgebras by Yau's twist principle. Also, we explore some general classes of BiHom-Lie admissible superalgebras and describe all these classes via $G$-BiHom-associative superalgebras, where $G$ is a subgroup of the symmetric group $S_{3}$. Finally, we discuss the concept of $β^{k}$-derivation…
The paper studies nilpotent $n$-Lie superalgebras. More specifically speaking, we first prove Engel's theorem for $n$-Lie superalgebras. Second, we research some properties of nilpotent $n$-Lie superalgebras, Finally, we give several sufficient conditions that an $n$-Lie superalgebra is nilpotent.
For a finitely generated integral super domain $A$, we prove the Lie superalgebra $\mathcal{V} = Der(A)$ of super derivations is a simple Lie superalgebra.
In this paper, we introduce the notion of n-Hom-pre-Lie superalgebras. We investigate the representation theory of n-Hom-pre-Lie superalgebras and we give some related results and structures based on Rota-Baxter operators, O-operators and Nijenhuis operators. Moreover, we study relationships between n-Hom-pre-Lie superalgebras and its induced Hom-pre-Lie superalgebras. The same procedure is applied for the representations of n- Hom-pre-Lie superalgebras.
We consider the $δ$-derivations of classical Lie superalgebras and prove that these superalgebras admit nonzero $δ$-derivations only when $δ= 0,1/2,1$. The structure of $1/2$-derivations for classical Lie superalgebras is completely determined.
In this paper, we construct free Lie Rota-Baxter superalgebra by using Gröbner-Shirshov bases theory. We firstly construct free operated Lie superalgebras by the operated super-Lyndon-Shirshov monomials. Secondly, we establish Gröbner-Shirshov bases theory for operated Lie superalgebras. Thirdly, we find a Gröbner-Shirshov basis of a free Lie Rota-Baxter superalgebra on a $\mathbb{Z}_2$-graded set. Consequently, we can obtain a linear basis of a free Lie Rota-Baxter superalgebra by the composition-diamond lemma for operated Lie superalgebras.
In this article, we define the capable pairs of Lie superalgebras. We classify all capable pairs of abelian and Heisenberg Lie superalgebras. After that we discuss on pairs of Lie superalgebras with derived subalgebra of dimension one and a non-abelian ideal. Finally, we determine the structure of the Schur multiplier of pairs of Heisenberg Lie superalgebras.
HNN-extension of Lie superalgebras
本文研究了李超代数的HNN扩张构造，并证明了有限维李超代数可以嵌入到双生成的李超代数中。
We generalize to the case of Lie superalgebras the classical symplectic double extension of symplectic Lie algebras introduced in [2]. We use this concept to give an inductive description of nilpotent homogeneous-symplectic Lie superalgebras. Several examples are included to show the existence of homogeneous quadratic symplectic Lie superalgebras other than even-quadratic even-symplectic considered in [6]. We study the structures of even (resp. odd)-quadratic odd (resp. even)-symplectic Lie superalgebras and odd-quadratic odd-symplectic Lie superalgebras and we give its inductive descriptions …
Models of the exceptional simple modular Lie superalgebras in characteristic $p\geq 3$, that have appeared in the classification due to Bouarroudj, Grozman and Leites of the Lie superalgebras with indecomposable symmetrizable Cartan matrices, are provided. The models relate these exceptional Lie superalgebras to some low dimensional nonassociative algebraic systems.
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Lie conformal superalgebras. Firstly, we construct the semidirect product of a Lie conformal superalgebra and its conformal module, and study derivations of this semidirect product. Secondly, we develop cohomology theory of Lie conformal superalgebras and discuss some applications to the study of deformations of Lie conformal superalgebras. Finally, we introduce generalized derivations of Lie conformal superalgebras and study their properties.
The aim of this note is to completely determine the second homology group of the special queer Lie superalgebra $\mathfrak{sq}_n(R)$ coordinatized by a unital associative superalgebra $R$, which will be achieved via an isomorphism between the special linear Lie superalgebra $\mathfrak{sl}_{n}(R\otimes Q_1)$ and the special queer Lie superalgebra $\mathfrak{sq}_n(R)$.
In this paper, we characterize the local superderivations on Cartan type Lie superalgebras over the complex field $\mathbb{C}$. Furthermore, we prove that every local superderivations on Cartan type simple Lie superalgebras is a superderivations. As an application, using the results on local superderivations we characterize the $2$-local superderivations on Cartan type Lie superalgebras. We prove that every $2$-local superderivations on Cartan type Lie superalgebras is a superderivations.
In this paper, we study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras, we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.
Representations of Color Lie Superalgebras by Hilbert Series
本文研究Color Lie超代数的Hilbert级数表示，提出了一个类似于自由Color Lie超代数和特定Color Lie p-超代数的Witt公式的维数公式。
In this article we define the $c$-nilpotent multiplier of a finite dimensional Lie suepralgebra. We characterize the structure of $2$-nilpotent multiplier of finite dimensional nilpotent Lie superalgebras whose derived subalgebras have dimension at most one. Then we give an upper bound on the dimension of $2$-nilpotent multiplier of any finite dimensional nilpotent Lie superalgebra. Moreover, we discuses the $2$-capability of special as well as odd Heisenberg Lie superalgebras and abelian Lie superalgebras.
In this paper we study the varieties of nilpotent Lie superalgebras of dimension $\leq 5$. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a by product we construct rigid nilpotent Lie superalgebras of arbitrary dimension.
In this paper, we characterize the super-biderivations of Cartan type Lie superalgebras over the complex field $\mathbb{C}$. Furthermore, we prove that all super-biderivations of Cartan type simple Lie superalgebras are inner super-biderivations.
Let $L$ be a Lie superalgebra over a field of characteristic different from $2,3$ and write $\mathrm{ID}^{*}(L)$ for the Lie superalgebra consisting of superderivations mapping $L$ to $L^{2}$ and the central elements to zero. In this paper we first give an upper bound for the superdimension of $\mathrm{ID}^{*}(L)$ by means of linear vector space decompositions. Then we characterize the $\mathrm{ID}^{*}$-superderivation superalgebras for the nilpotent Lie superalgebras of class 2 and the model filiform Lie superalgebras by methods of block matrices.
In this paper, we introduce a representation theory of Hom-Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular Hom-Lie conformal superalgebras. Finally, we introduce derivations of multiplicative Hom-Lie conformal superalgebras and study their properties.
On Hom-Lie algebras and superalgebras,we introduce the notions of biderivations, linear commuting maps and α-biderivations, and compute them for some typical Hom-Lie algebras and superalgebras, including q-deformed W(2,2) algebra, q-deformed Witt algebra and superalgebra.
Isoclinism of Lie superalgebras has been defined and studied currently. In this article it is shown that for finite dimensional Lie superalgebras of same dimension, the notation of isoclinism and isomorphism are equivalent. Furthermore we show that covers of finite dimensional Lie superalgebras are isomorphic using isoclinism concept.
HNN extensions of Lie superalgebras
本文明确刻画了李超代数HNN扩张的结构，证明其为原代数与自由李超代数的直和，并用于研究有限表示李超代数中理想的有限生成性，同时发展了Gröbner-Shirshov基理论。
Suppose that the underlying field is of characteristic different from $2, 3$. In this paper we first prove that the so-called stem deformations of a free presentations of a finite-dimensional Lie superalgebra $L$ exhaust all the maximal stem extensions of $L$, up to equivalence of extensions. Then we prove that multipliers and covers always exist for a Lie superalgebra and they are unique up to Lie superalgebra isomorphisms. Finally, we describe the multipliers, covers and maximal stem extensions of Heisenberg superalgebras of odd centers and model filiform Lie superalgebras.
Isoclinism and factor set in regular Hom-Lie Superalgebras
本文在正则Hom-Lie超代数中引入同源性与因子集概念，证明有限维同维数正则Hom-Lie超代数同源当且仅当同构。
Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.
In this paper we define isoclinism for Lie superalgebras and using the concept of isoclinism, we give the structure of all covers of Lie superalgebras when their Schur multipliers are finite dimensional. It has been shown that that maximal stem extensions of Lie superalgebras are precisely same as the stem covers. Furthermore, we have defined stem Lie superalgebra and prove that each isoclinic family $\mathcal{C}$ contain a stem Lie superalgebra $T$ and it is the one having minimum even and odd dimension. Finally we have proved a converse of Schur's theorem and have given bound for stem Lie su…
We introduce the non-abelian tensor product of Lie superalgebras, study some of its properties including nilpotency, solvability and Engel, and we use it to describe the universal central extensions of Lie superalgebras. We present the low-dimensional non-abelian homology of Lie superalgebras and establish its relationship with the cyclic homology of associative superalgebras. We also define the non-abelian exterior product and give an analogue of Miller's theorem, Hopf formula and a six-term exact sequence for the homology of Lie superalgebras.
On the dimension of the Schur multiplier of nilpotent Lie superalgebras
本研究针对有限维幂零李超代数的Schur乘子维数问题，提出了一个比已有李超代数Schur乘子维数界更精确的界，改进了相关结果。
In this paper, we give an expansion of two notions of double extension and $T^*$-extension for quadratic and odd quadratic Lie superalgebras. Also, we provide a classification of quadratic and odd quadratic Lie superalgebras up to dimension 6. This classification is considered up to isometric isomorphism, mainly in the solvable case and the obtained Lie superalgebras are indecomposable.
Recently, in [18] the authors gave some results on the structure, capability and the Schur multiplier of generalized Heisenberg Lie superalgebra. In this work we try to extend these concepts to the case of generalized Heisenberg Lie superalgebra.
We show that the universal central extension of a direct limit of perfect Lie superalgebras L_i is (isomorphic to) the direct limit of the universal central extensions of L_i. As an application we describe the universal central extensions of some infinite rank Lie superalgebras.
Many theorems and formulas of Lie algebras run quite parallel to Lie superalgebra case, sometimes giving interesting results. So it is quite natural to extend the new concepts of Lie algebra immediately to Lie superalgebra case, as these type of algebras have wide applications in physics and related theories. Using the concept of isoclinism, F. Saeedi and S. Sheikh-Mohseni recently studied the central derivation of nilpotent Lie algebra with nilindex 2. The purpose of the present paper is to continue and extend the investigation to obtain some similar results for Lie superalgebras, as isoclini…
Let L be a Lie superalgebra over a fled of characteristic p\neq 2 with enveloping algebra U(L) or let L be a restricted Lie superalgebra over a field of characteristic p > 2 with restricted enveloping algebra u(L). In this note, we establish when u(L) or U(L) is bounded Lie Engel.
On solvable Lie superalgebras of maximal rank
本文研究了李超代数的超导子性质，在给定幂零根的条件下估计了补空间维数上界，并证明了极大秩可解李超代数同构于极大秩幂零根的极大可解扩张。
In this paper, all (super)algebras are over a field $\mathbb{F}$ of characteristic different from $2, 3$. We construct the so-called 5-sequences of cohomology for central extensions of a Lie superalgebra and prove that they are exact. Then we prove that the multipliers of a Lie superalgebra are isomorphic to the second cohomology group with coefficients in the trivial module for the Lie superalgebra under consideration.
We give necessary conditions for the existence of degenerations between two complex Lie superalgebras of dimension $(m,n)$. As an application, we study the variety $\mathcal{LS}^{(2,2)}$ of complex Lie superalgebras of dimension $(2,2)$. First we give the algebraic classification and then obtain that $\mathcal{LS}^{(2,2)}$ is the union of seven irreducible components, three of which are the Zariski closures of rigid Lie superalgebras. As byproduct, we obtain an example of a nilpotent rigid Lie superalgebra, in contrast to the classical case where no example is known.
In this paper some characterizations of Hom-Leibniz superalgebras are given and some of their basic properties are found. These properties can be seen as a generalization of corresponding well-known properties of Hom-Leibniz algebras. Considering the Hom-Akivis superalgebra associated to a given Hom-Leibniz superalgebra, it is observed that the Hom-super Akivis identity leads to an additional property of Hom-Leibniz superalgebras, which in turn gives a necessary and sufficient condition for Hom-super Lie admissibility of Hom-Leibniz superalgebras. We show also that every (left) Hom-Leibniz sup…
The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.
The Solvabilizers and Solvable Graphs in Lie Superalgebras
本文研究了李超代数的可解化剂与可解图，探讨了其作为同构不变量的应用及可解性度量的定义。
We study codimension growth of infinite dimensional Lie superalgebras over an algebraically closed field of characteristic zero. We prove that if a Lie superalgebra $L$ is a Grassmann envelope of a finite dimensional simple Lie algebra then the PI-exponent of $L$ exists and it is a positive integer.
Capability of Nilpotent Lie Superalgebras of Small Dimension
该研究引入部分可容李超代数概念，并据此完成维数不超过五的可容幂零李超代数的完整分类。
In this paper, we introduce the concept of (super-)multiplier-rank for Lie superalgeras and classify all the finite-dimensional nilpotent Lie superalgebras of multiplier-rank $\leq 2$ over an algebraically closed field of characteristic zero. In the process, we also determine the multipliers of Heisenberg superalgebras.
This paper is devoted to investigating the structure theory of a class of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we completely determine the derivation algebras, the automorphism groups and the second cohomology groups of these Lie superalgebras.
Along this paper we show that under certain conditions the method for describing of solvable Lie and Leibniz algebras with maximal codimension of nilradical is also extensible to Lie and Leibniz superalgebras, respectively. In particular, we totally determine the solvable Lie and Leibniz superalgebras with maximal codimension of model filiform and model nilpotent nilradicals. Finally, it is established that the superderivations of the obtained superalgebras are inner.
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