M. V. Tvalavadze
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.
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Homogeneous O-operators on BiHom-Jordan superalgebras and BiHom-pre-Jordan superalgebras structures
该研究围绕BiHom-若尔当超代数展开，回顾其概念并研究表示与对偶表示理论，引入齐次O-算子概念，定义BiHom-预若尔当超代数结构，揭示二者通过齐次O-算子的关联及相关结果。
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.
I.P. Shestakov constructed an example of a unital simple special Jordan superalgebra over the field of real numbers. It turned out to be a subsuperalgebra of the Jordan superalgebra of vector type, but not isomorphic to a superalgebra of this type. Moreover, its superalgebra of fractions is isomorphic to a Jordan superalgebra of vector type. Author constructed a similar example of a Jordan superalgebra over a field of characteristic~0 in which the equation $t^2+1=0$. In this article we present an~example of a Jordan superalgebra with the same properties over an arbitrary field of characteristi…
We construct two superalgebras associated to a punctured Riemann surface. One of them is a Lie superalgebra of Krichever-Novikov type, the other one is a Jordan superalgebra. The constructed algebras are related in several ways (algebraic, geometric, representation theoretic). In particular, the Lie superalgebra is the algebra of derivations of the Jordan superalgebra.
In this article we begin the study of representations of simple finite-dimensional noncommutative Jordan superalgebras. In the case of degree $\geq 3$ we show that any finite-dimensional representation is completely reducible and, depending on the superalgebra, quasiassociative or Jordan. Then we study representations of superalgebras $D_t(α,β,γ)$ and $K_3(α, β, γ)$ and prove the Kronecker factorization theorem for superalgebras $D_t(α,β,γ)$. In the last section we use a new approach to study noncommutative Jordan representations of simple Jordan superalgebras.
In this paper we describe all subalgebras and automorphisms of simple noncommutative Jordan superalgebras $K_3(α,β,γ)$ and $D_t(α,β,γ)$ and compute the derivations of the nontrivial simple finite-dimensional noncommutative Jordan superalgebras.
The group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras.
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.
We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.
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.
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 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.
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.
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。
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…
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。
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…
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。
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。
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…
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.
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.
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.
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.}
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…
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.
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。
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。
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。
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。
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.
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…
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。
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。
We classify simple finite Jordan conformal superalgebras and establish preliminary results for the classification of simple finite Jordan pseudoalgebras.
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…
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。
We described $δ$-derivations and $δ$-superderivations of simple and semisimple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. We constructed new examples of 1/2-derivations and 1/2-superderivations of simple Zelmanov's superalgebra $V_{1/2}(Z,D).$
We introduce the concept of a $δ$-superderivation of a superalgebra. $δ$-Derivations of Cartan-type Lie superalgebras are treated, as well as $δ$-superderivations of simple finite-dimensional Lie superalgebras and Jordan superalgebras over an algebraically closed field of characteristic 0. We give a complete description of $\{1}{2}$-derivations for Cartan-type Lie superalgebras. It is proved that nontrivial $δ$-(super)derivations are missing on the given classes of superalgebras, and as a consequence, $δ$-superderivations are shown to be trivial on simple finite-dimensional noncommutative Jord…
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.
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.
$δ$-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 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。
We described $δ$-derivations and $δ$-superderivations of simple Jordan superalgebra <<KKM Double>> (also known as superalgebra of Jordan brackets) and unital simple finite-dimensional Jordan superalgebras over algebraic closed fields with characteristic $p\neq2$. As a consequence, we received relationship between non-trivial $δ$-derivations and specialty of simple superalgebra <<KKM Double>>. We constructed new examples of non-trivial 1/2-derivations of Jordan superalgebras of vector type.
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.
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。
We investigate an analogue to the Wedderburn Principal Theorem (WPT) for a finite-dimensional Jordan superalgebra $J$ with solvable radical $N$ such that $N^2=0$ and $J/N\cong JP_n$, $n\geq 3$. We consider $N$ as an irreducible $JP_n$-bimodule and we prove that the WPT holds for $J$.
Engel's Theorem for Alternative Superalgebras
本文在无限域上Jordan超代数Engel定理基础上，为有限维交替超代数给出nilpotency判据，无需限制基域基数，并探讨了graded-nil与nilpotent交替超代数的联系，给出了非nilpotent的Engelian可交换幂结合超代数例。
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 $\mathfrak{osp}(1|2)$.
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.
In this note we mainly study the fine Jordan-Chevalley decomposition: a refinement of the classical Jordan-Chevalley decomposition of a matrix and we pay a particular attention to the field of the coefficients of the matrix. Moreover we obtain some further additive and multiplicative decompositions of a matrix under suitable conditions.
We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic linear operators, which are related by the exponential map. We also give a new proof of the theorem of Jordan canonical form for quaternionic linear operators, which is intrinsic and takes less computations than the known proofs for square quaternionic matrices.
In this paper we consider Lie superalgebras decomposable as the sum of two proper subalgebras. Any of these algebras has the form of the vector space sum $L=A+B$ where $A$ and $B$ are proper simple subalgebras which need not be ideals of $L$, and the sum need not be direct. The main result of this paper is the following: Let $S = {osp}(m,2n)$ be a Lie superalgebra such that $S=K+L$ where $K$, $L$ are two proper basic simple subalgebras. Then $m$ is even, $m=2k$ and $K \cong osp(2k-1,2n)$, $L \cong sl(k,n)$.
A tripotent of a generalized triple system of second order defines a decomposition of the space of the space which consist of 10 components. It gives a generalization of the Peirce decomposition n for the Jordan triple system which consists in general of 4 components.
Decomposition algebras and axial decomposition algebras are classes of commutative nonassociative algebras which are generalizations of axial algebras. The classes decomposition algebras, axial decomposition al;gebras and non-primitive axial algebras also have univesal algebras. Furthermore, by using the existence of a universal algebra, 2-generated non-primitive axial algebras of Jordan type is classified.
The purpose of this note is to advertise an elegant algorithmic proof for the Jordan--Chevalley decomposition of a matrix, following and (slightly) revising the discussion of Couty, Esterle und Zarouf (2011). The basic idea of that method goes back to Chevalley (1951).
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.
This paper presents some results on simple exceptional Jordan algebra over algebraically closed field $Φ$ with characteristic not 2. Namely an example of simple decomposition of $H(O_3)$ into the sum of two subalgebras of the type $H(Q_3)$ is produced, and it is shown that this decomposition is the only possible in terms of the type of simple subalgebras.
Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's Theorem about the characterization of the existence of characteristic non hyperinvariant subspaces.
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