The Kac Jordan superalgebra: automorphisms and maximal subalgebras

Ternary superderivations of Jordan superalgebras

We describe the ternary and the generalized superderivations of finite-dimensional semisimple Jordan superalgebras over an algebraically closed field of characteristic zero and of finite-dimensional simple Jordan superalgebras with semisimple even part over an algebraically closed field of any characteristic not equal 2.

Simple decompositions of simple special Jordan superalgebras

We classify decompositions of simple special finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic zero into the sum of two proper simple subsuperalgebras.

Degenerations of Jordan Superalgebras

We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.

Classification of three-dimensional Jordan superalgebras

In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer to the problem of determining the minimal dimension of exceptional Jordan superalgebras. Our strategy to provide a lower bounded for this dimension is to determine the complete list of Jordan superalgebras of small dimensions and verify which ones are special or exceptional.

Second cohomology group of the finite-dimensional simple Jordan superalgebra $\mathcal{D}_{t}$, $t\neq 0$

The second cohomology group (SCG) of the Jordan superalgebra $\mathcal{D}_{t}$, $t\neq 0$, is calculated by using the coefficients which appear in the regular superbimodule $\mathrm{Reg}\mathcal{D}_t$. Contrary to the case of algebras, this group is nontrivial thanks to the non-splitting caused by the Wedderburn Decomposition Theorem \cite{Faber1}. First, to calculate the SCG of a Jordan superalgebra we use split-null extension of the Jordan superalgebra and the Jordan superalgebra representation. We prove conditions that satisfy the bilinear forms $h$ that determine the SCG in Jordan superalg…

Hom-Alternative, Hom-Malcev and Hom-Jordan superalgebras

Hom-alternative, Hom-Malcev and Hom-Jordan superalgebras are $\mathbb{Z}_{2}$-graded generalizations of Hom-alternative, Hom-Malcev and Hom-Jordan algebras, which are Hom-type generalizations of alternative, Malcev and Jordan algebras. In this paper we prove that Hom-alternative superalgebras are Hom-Malcev-admissible and are also Hom-Jordan-admissible. Home-type generalizations of some well known identities in alternative superalgebras, including the $\mathbb{Z}_{2}$-graded Bruck-Kleinfled function are obtained.

Maximal subalgebras of Jordan superalgebras

The maximal subalgebras of the finite dimensional simple special Jordan superalgebras over an algebraically closed field of characteristic 0 are studied. This is a continuation of a previous paper by the same authors about maximal subalgebras of simple associative superalgebras, which is instrumental here.

简单单值Jordan超代数

Simple unital Jordan superalgebras

本文证明任意维数的简单单值Jordan超代数要么属于已知简单单值超代数列表，要么位于某个真子簇中。

On Jordan doubles of slow growth of Lie superalgebras

To an arbitrary Lie superalgebra $L$ we associate its Jordan double ${\mathcal Jor}(L)$, which is a Jordan superalgebra. This notion was introduced by the second author before. Now we study further applications of this construction. First, we show that the Gelfand-Kirillov dimension of a Jordan superalgebra can be an arbitrary number $\{0\}\cup [1,+\infty]$. Thus, unlike associative and Jordan algebras, one hasn't an analogue of Bergman's gap $(1,2)$ for the Gelfand-Kirillov dimension of Jordan superalgebras. Second, using the Lie superalgebra $\mathbf R$ constructed before, we construct a Jor…

Jordan 超代数的代数与几何分类

The algebraic and geometric classification of Jordan superalgebras

本文对复数域上四维 Jordan 超代数进行了代数与几何分类，并刻画了相应簇的不可约分支。

On structure and TKK algebras for Jordan superalgebras

We compare a number of different definitions of structure algebras and TKK constructions for Jordan (super)algebras appearing in the literature. We demonstrate that, for unital superalgebras, all the definitions of the structure algebra and the TKK constructions fall apart into two cases. Moreover, one can be obtained as the Lie superalgebra of superderivations of the other. We also show that, for non-unital superalgebras, more definitions become non-equivalent. As an application, we obtain the corresponding Lie superalgebras for all simple finite dimensional Jordan superalgebras over an algeb…

On Kac's Jordan superalgebra

The group-scheme of automorphisms of the ten-dimensional exceptional Kac's Jordan superalgebra is shown to be isomorphic to the semidirect product of the direct product of two copies of SL2 by the constant group scheme C2. This is used to revisit, extend, and simplify, known results on the classification of the twisted forms of this superalgebra and of its gradings.

具有Clifford型偶部的单Jordan超代数

Simple Jordan superalgebras with the even parts of Clifford type

本文研究单无限维Jordan超代数的分类，证明偶部有限维时仅有超形式超代数，并进一步证明偶部为Clifford型时直和项数只能为1，且此时仍为超形式超代数。

非交换约当超代数的代数与几何分类

The algebraic and geometric classification of noncommutative Jordan superalgebras

本文对复数3维非交换约当超代数及其相关超代数进行了代数与几何分类，揭示了其结构特征与分类方法。

Wedderburn principal theorem for Jordan superalgebras I

We consider finite dimensional Jordan superalgebras $\jor$ over an algebraically closed field of characteristic 0, with solvable radical $\rad$ such that $\radd=0$ and $\jor/\rad$ is a simple Jordan superalgebra of one of the following types: Kac $\kac$, Kaplansky $\mathcal{K}_3$ superform or $\algdt$. We prove that an analogue of the Wedderburn Principal Theorem (WPT) holds if certain restrictions on the types of irreducible subsuperbimodules of $N$ are imposed, where $N$ is considered as a $J$-superbimodule, and $J$ is a simple Jordan superalgebra. Using counterexamples, it is shown that the…

Algebras of Jordan brackets and Generalized Poisson algebras

We construct a basis of free unital generalized Poisson superalgebras and a basis of free unital superalgebras of Jordan brackets. Also, we prove the analogue of Farkas' Theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations beetwen generic Poisson superalgebras and superalgebras of Jordan brackets will be studied.

The Supermagic Square in characteristic 3 and Jordan superalgebras

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.

Irreductible Representations of the Simple Jordan Superalgebra of Grassmann Poisson bracket

We obtain classification of the irreducible bimodules over the Jordan superalgebra $Kan(n)$, the Kantor double of the Grassmann Poisson superalgebra $G_n$ on $n$ odd generators, for all $n \geq 2$ and an algebraically closed field of characteristic $\ne 2$. This generalizes the corresponding result of C.Martínez and E.Zelmanov announced in \cite{MZ2} for $n>4$ and a field of characteristic zero.}

Prime (-1,1) and Jordan monsters and superalgebras of vector type

It is proved that the prime degenerate (-1,1) algebra constructed in [13] (the (-1,1)-monster) generates the same variety of algebras as the Grassman (-1,1)-algebra. Moreover, the same variety is generated by the Grassmann envelope of any simple nonassociative (-1,1)-superalgebra. The variety occurs to be the smallest variety of (-1,1)-algebras that contains prime nonassociative algebras. Similar results are obtained for Jordan algebras. Thus, the Jordan monster (the prime degenerate algebra constructed in [13]) and the Grassmann envelope of the prime Jordan superalgebra of vector type have th…

Group gradings on the Lie and Jordan superalgebras $Q(n)$

We classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras $Q(n)$, $n \geq 2$, over an algebraically closed field of characteristic different from $2$ (and not dividing $n+1$ in the Lie case): fine gradings up to equivalence and $G$-gradings, for a fixed group $G$, up to isomorphism.

类型 $(1,3)$ 和 $(3,1)$ 的 Jordan 超代数簇的不可约分支

Irreducible Components of the Varieties of Jordan Superalgebras of Types $(1,3)$ and $(3,1)$

本文描述了偶部分维数为 $1$ 或 $3$ 的四维 Jordan 超代数簇，证明其由 $11$ 和 $21$ 个刚性超代数的轨道 Zariski 闭包并集构成，并给出不可约分支，同时通过构造四维可解刚性超代数否定了 Vergne 猜想的类比。

特征5下通过张量范畴从Albert代数到Kac十维Jordan超代数

From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5

本研究在特征5域下通过张量范畴半单化，由Albert代数与5阶自同构得到Kac十维Jordan超代数，解释了相关奇特结果，还得到特征5特有李超代数el(5;5)并给出超代数构造的一般方法。

维数为四且偶部分维数为二的Jordan超代数的簇

The Variety of Jordan Superalgebras of dimension four and even part of dimension two

本文描述了维数为四且偶部分维数为二的Jordan超代数的簇，证明其有25个不可约分支，并构造性地解决了第二上同调群不消没的刚性Jordan超代数的存在性问题。

nilpotent Jordan superalgebras 的五维变种

Varieties of nilpotent Jordan superalgebras of dimension five

本文研究五维复nilpotent Jordan超代数的变种，通过Jordan标准形和矩阵三角化等方法确定其同构类，并提供几何分类，描述变种间的退化与非退化。

Orthosymplectic Jordan superalgebras and the Wedderburn principal theorem (WPT)

An analogue of the Wedderbur principal theorem (WPT) is considered for finite dimensional Jordan superalgebras A with solvable radical N, such that N^2=0 and A/N is isomorphic to Josp_n|2m(F), where F is an algebraicallly closed field of characteristic zero. Let's we prove that the WPT is valid under some restrictions over the irreducible Josp_n|2m(F)-bimodules contained in N, and it is shown with counter-examples that these restrictions can not be weakened.

Supercommutator (Hom-)superalgebras of right (Hom-)alternative superalgebras

It shown that the supercommutator superalgebra of a right alternative superalgebra is a Bol superalgebra. Hom-Bol superalgebras are defined and it is shown that they are closed under even self-morphisms. Any Bol superalgebra along with any even self-morphism is twisted into a Hom-Bol superalgebra. The supercommutator superalgebra of a right Hom-alternative superalgebra has a natural Hom-Bol structure. In order to prove this last result, the Hom-Jordan-admissibility of right Hom-alternative superalgebras is investigated and next Hom-Jordan supertriple systems are defined and their connection wi…

关于单的转置泊松代数与若尔当超代数

On the simple transposed Poisson algebras and Jordan superalgebras

本文证明了转置泊松代数的单性等价于其关联李括号的单性，并由此推出特征零代数闭域上有限维单转置泊松代数平凡，同时将结论推广到超代数情形，并给出其 Kantor double 的若尔当超代数性质与单性判据。

对约当超代数的变种的不可约分量的完整分类

The complete classification of irreducible components of varieties of Jordan superalgebras

本文对约当超代数类型(3,1)和(2,2)的变种进行了不可约分量的完整分类，回答了相关开放问题。

Irreducible representations of the exceptional cheng-kac superalgebra

We classify all conformal irreducible modules of finite type over the Cheng Kac superalgebra CK(6).

Simple Jordan conformal superalgebras

We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.

Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras

We construct a just infinite fractal 3-generated Lie superalgebra $\mathbf Q$ over arbitrary field, which gives rise to an associative hull $\mathbf A$, a Poisson superalgebra $\mathbf P$, and two Jordan superalgebras $\mathbf J$, $\mathbf K$. One has a natural filtration for $\mathbf A$ which associated graded algebra has a structure of a Poisson superalgebra and $\mathrm{gr} \mathbf A\cong\mathbf P$, also $\mathbf P$ admits an algebraic quantization. The Lie superalgebra $\mathbf Q$ is finely $\mathbb Z^3$-graded by multidegree in the generators, $\mathbf A$, $\mathbf P$ are $\mathbb Z^3$-gr…

Gradings on the Kac superalgebra

We classify the gradings on the Kac superalgebra up to equivalence.

Jordan超代数的循环同调及其相关Lie超代数

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。

分级嵌入、根生成子代数与 $π$-系统在拟简单Kac-Moody超代数中的应用

Graded embeddings, root generated subalgebras and $π$-systems for quasisimple Kac-Moody superalgebras

本文研究了拟简单Kac-Moody超代数中根生成子代数与$π$-系统的关系，通过反射操作获得实根并建立Dynkin双射的类比，同时探讨了超设置中根字符串的性质。

半单有限维Jordan超代数的δ-超导子

$δ$-Superderivations of semisimple finite-dimensional Jordan superalgebras

本文刻画了特征p≠2的代数闭域上半单有限维Jordan超代数的δ-导子与δ-超导子，给出了非平凡的1/2-导子及奇数1/2-超导子的新实例，这些实例不属于超代数元素右乘算子。

单有限维Jordan与李超代数的δ-超导子

$δ$-superderivations of simple finite-dimensional Jordan and Lie superalgebras

该研究引入超代数的δ-超导子概念，刻画Cartan型李超代数的{1}/{2}-导子，证明特征0代数闭域上相关超代数的δ-(超)导子均为平凡，并研究了特定Jordan代数的δ-导子。

The extended Freudenthal Magic Square and Jordan algebras

The Lie superalgebras in the extended Freudenthal Magic Square in characteristic 3 are shown to be related to some known simple Lie superalgebras, specific to this characteristic, constructed in terms of orthogonal and symplectic triple systems, which are defined in terms of central simple degree three Jordan algebras.

On Kac-Weisfeiler modules for general and special linear Lie superalgebras

Let $\ggg:=\gl_{m|n}$ be a general linear Lie superalgebra over an algebraically closed field $\mathds{k}=\overline{\mathbb{F}}_p$ of characteristic $p>2$. A module of $\ggg$ is said to be of Kac-Weisfeiler if its dimension coincides with the dimensional lower bound in the super Kac-Weisfeiler property presented by Wang-Zhao in \cite{WZ}. In this paper, we verify the existence of the Kac-Weisfeiler modules for $\gl_{m|n}$. We also establish the corresponding consequence for the special linear Lie superalgebras $\mathfrak{sl}_{m|n}$ with restrictions $p>2$ and $p\nmid(m-n)$.

Some results on $δ$-Hom-Jordan Lie conformal superalgebras

In this paper, we introduce the representation theory of $δ$-Hom-Jordan Lie conformal superalgebras and discuss the cases of adjoint representations. Furthermore, we develop the cohomology theory of Hom-Lie conformal superalgebras and discuss some applications to the study of deformations of regular $δ$-Hom-Jordan Lie conformal superalgebras. Finally, we introduce derivations of multiplicative $δ$-Hom-Jordan Lie conformal superalgebras and study their properties.

单有限维Jordan超代数的$\delta$-导子

$δ$-derivations of simple finite-dimensional Jordan superalgebras

该研究刻画了两类代数结构的非平凡$\delta$-导子，证明了特定$\delta$值外无对应非零非平凡导子，并完整给出了$\frac{1}{2}$-导子的相关结论。

Yang-Baxter方程与带表示的Hom-Jordan超代数的$\mathcal O$-算子

Yang-Baxter equations and $\mathcal O$-operators of a Hom-Jordan superalgebra with representation

本文研究Hom-Jordan超代数上的Yang-Baxter方程，建立$\mathcal{O}$-算子与斜对称解的联系，并探讨Hom-pre-Jordan超代数与Hom-Jordan超代数之间的关系。

关于Jordan括号的单超代数的$\delta$-超导子

On $δ$-superderivations of simple superalgebras of Jordan brackets

本文刻画了两类Jordan超代数的$\delta$-导子与$\delta$-超导子，进而得到单单位Jordan括号超代数为特殊型的判定准则。

Group gradings on superalgebras

In this paper we look into the structure of finite-dimensional graded superalgebras of various types such as associative, Lie and Jordan over an algebraically closed field of characteristic zero.

有限维超代数中的中心序环

Central orders in simple finite dimensional superalgebras

本文研究了有限维超代数中中心序环的性质，探讨了关联、经典约尔旦及部分替代超代数中的相关定理。

On the double of the (restricted) super Jordan plane

We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same authors. We show that their Drinfeld doubles give rise naturally to Hopf superalgebras justifying a posteriori the adjective super given in their work. These Hopf superalgebras are extensions of super commutative ones by the enveloping, respectively restricted enveloping, algebra of $\mathfrak{osp}(1|2)$.

Generalized conformal realizations of Kac-Moody algebras

We present a construction which associates an infinite sequence of Kac-Moody algebras, labeled by a positive integer n, to one single Jordan algebra. For n=1, this reduces to the well known Kantor-Koecher-Tits construction. Our generalization utilizes a new relation between different generalized Jordan triple systems, together with their known connections to Jordan and Lie algebras. Applied to the Jordan algebra of hermitian 3x3 matrices over the division algebras R, C, H, O, the construction gives the exceptional Lie algebras f4, e6, e7, e8 for n=2. Moreover, we obtain their infinite-dimensio…

The minimal number of generators for simple Lie superalgebras

Using the classification theorem due to Kac we prove that any finite dimensional simple Lie superalgebra over an algebraically closed field of characteristic 0 is generated by one element.

Generalized δ-Derivations

We defined generalized δ-derivations of algebra A as linear mapping χassociated with usual δ-derivation ϕby the rule χ(xy)=δ(χ(x)y+xϕ(y))=δ(ϕ(x)y+xχ(y)) for any x,y \in A. We described generalized δ-derivations of prime alternative algebras, prime Lie algebras and superalgebras, unital algebras, and semisimple finite-dimensional Jordan superalgebras. In this cases we proved that generalized δ-derivation is a generalized derivation or δ-derivation. After that we described δ-superderivations of superalgebras <<KKM Double>>, arising from prime alternative algebras, prime Lie algebras and superalg…

几乎单的代数超群

Almost-simple algebraic supergroups

本文研究了几乎单的代数超群，构造了新的超群，其李超代数不在Kac定理列表中。