Susanne Pumpluen
We construct Jordan algebras over a locally ringed space using generalizations of the Tits process and the first Tits construction by Achhammer. Some general results on the structure of these algebras are obtained. Examples of Albert algebras over a Brauer-Severi variety with associated central simple algebra of degree 3 are given.
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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。
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 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。
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。
For any $n\geq 3$, we prove that there exist equivalences between these apparently unrelated objects: irreducible $n$-dimensional non degenerate projective varieties $X\subset \mathbb P^{2n+1}$ different from rational normal scrolls and 3-covered by twisted cubic curves, up to projective equivalence; quadro-quadric Cremona transformations of $ \mathbb P^{n-1}$, up to linear equivalence; $n$-dimensional complex Jordan algebras of rank three, up to isotopy. We also provide some applications to the classification of particular classes of varieties in the class defined above and of quadro-quadric …
We describe degenerations of three-dimensional Jordan superalgebras over $\mathbb{C}$. In particular, we describe all irreducible components in the corresponding varieties.
The classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over algebraically closed fields and $\mathbb{R}$ is presented in terms of their matrices of structure constants.
We give algebraic and geometric classifications of complex $4$-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only $18$ non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of $3$ rigid algebras and $2$ one-parametric families of algebras.
The fundamental identity of quadratic Jordan algebras $Q_{Q_a b} = Q_aQ_bQ_a$ is commonly proven as a consequence of MacDonalds theorem or using more analytic methods. In this short note we give a self-contained purely algebraic proof using just a few easily proven identities and a Python script that follows a simple randomised logic to reduce expressions of Jordan operators.
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。
The algebraic and geometric classification of noncommutative Jordan algebras
提出由同维 Jordan 代数分类推导非交换 Jordan 代数分类的方法，完成复数域三维情形及多类相关代数的分类，并刻画相应代数簇的不可约分支。
The algebraic and geometric classification of derived Jordan and bicommutative algebras
本文提出新方法分类n维导出约尔旦代数，并应用于3维情况，同时得到3维metabelian交换代数和导出交换结合代数的代数分类，进而分类双交换代数，并探讨3维metabelian交换、导出交换结合、导出约尔旦和双交换代数的几何分类。
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.
In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.
In this paper, we classify four-dimensional Jordan algebras over an algebraically closed field of characteristic different of two. We establish the list of 73 non-isomorphic Jordan algebras.
For a given Jacobi-Jordan algebra $A$ and a vector space $V$ over a field $k$, a non-abelian cohomological type object ${\mathcal H}^{2}_{A} \, (V, \, A)$ is constructed: it classifies all Jacobi-Jordan algebras containing $A$ as a subalgebra of codimension equal to ${\rm dim}_k (V)$. Any such algebra is isomorphic to a so-called \emph{unified product} $A \, \natural \, V$. Furthermore, we introduce the bicrossed (semi-direct, crossed, or skew crossed) product $A \bowtie V$ associated to two Jacobi-Jordan algebras as a special case of the unified product. Several examples and applications are …
We announce here a number of results concerning representation theory of the algebra $R=k<x,y>/ (xy-yx-y^2)$, known as Jordan plane (or Jordan algebra). We consider the question on 'classification' of finite-dimensional modules over the Jordan algebra. Complete description of irreducible components of the representation variety $mod (R,n)$, which we call a Jordan variety' is given for any dimension $n$. It is obtained on the basis of the stratification of this variety related to the Jordan normal form of $Y$. Any irreducible component of the representation variety contains only one stratum rel…
A variety of associative algebras is called nonmatrix if it does not contain the algebra of 2 x 2 matrices over the given field. Nonmatrix varieties were introduced and studied by V.N.Latyshev in relation with the Specht problem. Some characterizations of nonmatrix varieties were obtained in the paper [10]. In the given paper the notion of nonmatrix variety is extended for nonassociative algebras, and the characterization from [10] is generalized for alternative, Jordan, and some other varieties of algebras.
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions $n<7$ over algebraically closed fields of characteristic not $2$ or $3$.
We give a computional method to construct and classify nilpotent Jordan algebras over any arbitrary fields by the second cohomolgy of nilpotent Jordan algebras of low dimension "analogue of Skjelbred-Sund method", we see that every nilpotent Jordan algebras can be constructed by the second cohomolgy of nilpotent Jordan algebras of low dimension. We use this method to classify nilpotent Jordan algebras up to dimension three over any field and nilpotent Jordan algebras of dimension four over an algebraic closed field of characteristic not 2 and over the real field R. Also commutative nilpotent a…
Invariant subspaces of a matrix $A$ are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of $A$. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.
Let $\boldsymbolΛ_3(\mathbb C)\,(=\mathbb C^{27})$ be the space of structure vectors of $3$-dimensional algebras over $\mathbb C$ considered as a $G$-module via the action of $G={\rm GL}(3,\mathbb C)$ on $\boldsymbolΛ_3(\mathbb C)$ `by change of basis'. We determine the complete degeneration picture inside the algebraic subset $\mathcal A^s_3$ of $\boldsymbolΛ_3(\mathbb C)$ consisting of associative algebra structures via the corresponding information on the algebraic subsets $\mathcal L_3$ and $\mathcal J_3$ of $\boldsymbolΛ_3(\mathbb C)$ of Lie and Jordan algebra structures respectively. Thi…
We classify up to isomorphism all gradings by an arbitrary group $G$ on the Lie algebras of zero-trace upper block-triangular matrices over an algebraically closed field of characteristic $0$. It turns out that the support of such a grading always generates an abelian subgroup of $G$. Assuming that $G$ is abelian, our technique also works to obtain the classification of $G$-gradings on the upper block-triangular matrices as an associative algebra, over any algebraically closed field. These gradings were originally described by A. Valenti and M. Zaicev in 2012 (assuming characteristic $0$ and $…
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.
Using geometric approach we formulate quantum theory in terms of Jordan algebras. We analyze the notion of (quasi)particle (=elementary excitation of translation-invariant stationary state) and the scattering of (quasi)particles in this framework.
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.
Cohomologies and linear deformations of pre-Jacobi-Jordan algebras
本文引入左预Jacobi-Jordan代数的上同调理论，利用上同调方法研究其线性形变，并证明Nijenhuis算子生成的形变是平凡的。
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.
Key varieties for prime $\mathbb{Q}$-Fano threefolds defined by Jordan algebras of cubic forms. Part II
本文通过用 9 个和 10 个参数坐标化三次形式的 9 维非退化二次 Jordan 代数，给出与 $\mathbb{P}^{2}\times\mathbb{P}^{2}$ 纤维化相关的 14 维和 15 维关键仿射簇的新构造。
We introduce a pre-Jacobi-Jordan algebras and study some relevant properties such as bimodules, matched pairs. Besides, we established a pre-Jacobi-Jordan algebra built as a direct sum of a given pre-Jacobi-Jordan algebra $(\A, \cdot)$ and its dual $(\A^{\ast}, \circ),$ endowed with a non-degenerate symmetric bilinear form $B,$ where $\cdot$ and $\circ$ are the products defined on $\A$ and $\A^{\ast},$ respectively. Finally, after pre-Jacobi-Jordan algebras classification in dimension two, we thoroughly give some double constructions of pre-Jacobi-Jordan algebraic structures.
We review work classifying the physically distinct forms of 3-qubit entanglement using the elegant framework of Jordan algebras, Freudenthal triple systems and groups of type E_7. While this framework is, in the first instance, specific to three qubits, it is shown here how the essential features may be naturally generalised to an arbitrary number of qubits.
The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-Pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and $\mathcal{O}$-operators introduced in this paper. Hom-Pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the ot…
The paper is devoted to classify nilpotent Jordan algebras of dimension up to five over an algebraically closed field of characteristic not 2. We obtained a list of 35 isolated non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras and 6 families of non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras depending either on one or two parameters over an algebraically closed field of characteristic not 2 or 3. In addition to these algebras we obtained two non-isomorphic 5-dimensional nilpotent non-associative Jordan algebras over an algebraically closed field of c…
An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies certain fusion rules $Φ(\frac{1}{2})$. On the other hand, $3$-transposition groups $(G,D)$ can be algebraically characterised as Matsuo algebras $M_α(G,D)$ with idempotents satisfying the fusion rules $Φ(α)$ for some $α$. We classify the Jordan algebras $J$ which are isomorphic to a Matsuo algebra $M_{1/2}(G,D)$, in which case $(G,D)$ is a subgroup of the (algebraic) automorphism group of $J$; the only possibilities are $G = \operatorname{Sym}(n)$ and $G = 3^2:…
$δ$-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。
$δ$-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。
We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.
Finite basis problem for varieties of algebraic systems
本文综述了代数系统变种中有限基数问题，通过举例说明非有限基数变种及具有有限基数性质的变种，重点关注半群、群等代数结构。
We investigate relations between the properties of an algebra and its varieties of finite-dimensional module structures, on the example of the Jordan plane $R=k<x,y>/ (xy-yx-y^2)$. Complete description of irreducible components of the representation variety $mod (R,n)$ obtained for any dimension $n$, it is shown that the variety is equidimensional. The influence of the property of the non-commutative Koszul (Golod-Shafarevich) complex to be a DG-algebra resolution of an algebra (NCCI), on the structure of representation spaces is studied. It is shown that the Jordan plane provides a new exampl…
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