M. V. Tvalavadze
In this paper we determine all types and the canonical forms of simple subalgebras for each type of simple Jordan algebras and the number of conjugate classes corresponding to the given simple Jordan algebra.
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The group of automorphisms of the Kac Jordan superalgebra is described, and used to classify the maximal subalgebras.
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.
Characterizing Jordan embeddings between block upper-triangular subalgebras via preserving properties
本文刻画了复矩阵代数中分块上三角子代数之间的 Jordan 嵌入，给出其标准形式与嵌入判据，并证明在 n≥3 时 Jordan 嵌入等价于保持交换性和谱的连续单射。
The purpose of this paper is to continue the study of chief factors of a Lie algebra and to prove a further strengthening of the Jordan-Hölder Theorem for chief series.
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.
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.
Solid lines in axial algebras of Jordan type $\tfrac{1}{2}$ and Jordan algebras
本文证明Jordan型1/2原始轴代数为Jordan代数当且仅当其所有二元生成子代数都是实心的，并刻画了实心性与结合子为导子的等价条件及非实心情形的轴数量。
We prove that assosymmetric algebras under Jordan product are Lie triple. A Lie triple algebra is called special if it is isomorphic to a subalgebra of some plus-assosymmetric algebra. We establish that Glennie identitiy is valid for special Lie triple algebras, but not for all Lie triple algebras.
Solid subalgebras in algebras of Jordan type half
该研究引入Jordan型代数的实心2-生成子代数概念，证明一般2-生成子代数为实心，特征零时例外值也成立，由此推出特征零下Jordan型代数要么有无穷多轴，要么是Matsuo代数或其商，支撑了相关分类猜想。
We are concerned with the question when Hom-Lie structures on a Lie algebra are closed with respect to the Jordan product. Somewhat unexpectedly, this leads us to certain questions connected with the Yang-Baxter equation, and with decomposition of a Lie algebra into the sum of subalgebras with given properties.
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.
Jordan homomorphisms of triangular algebras over noncommutative algebras
本文将Benkovič关于交换环上三角矩阵代数Jordan同态的刻画推广到非交换环，并去除加法挠假设，证明了T_n(R)的Jordan同态限制在子代数T_n^0(R)上必为标准形式。
In this paper we study Cartan subalgebras in general and special linear algebras over a field of positive characteristic. We determined the conjugacy classes of Cartan subalgebras under the general linear groups, and count the explicit number of all Cartan subalgebras from its conjugacy when the base field is an arbitrary finite field.
Automorphism group schemes of special simple Jordan pairs of types I and IV
本文确定了类型I和IV的有限维简单Jordan对及其相关Jordan三元系统的自同构群概形，基域特征非2。
An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras
本文将 Petek-Šemrl 关于保持交换性与谱的映射刻画推广到结构矩阵代数，给出使此类单射连续映射必为 Jordan 嵌入的充要条件。
Just-infinite Jordan Banach algebras
本文研究刚好无限JB代数的结构，证明其要么是无限维旋子因子，要么与C*代数相关联。
Let $J$ be a finite-dimensional semi-simple Jordan algebra over an algebraically closed field of characteristic $0$. In this article, the diagonal action of the automorphism group of $J$ on the $n$-fold product $J\times\ldots \times J$ is studied. In particular, it is shown that the orbit through an $n$-tuple $x=(x_1,,\ldots,x_n)$ is closed if and only if the Jordan subalgebra generated by the elements $x_1,\ldots, x_n$ is semi-simple.
Let $K$ be a 2-torsion free ring with identity and $R_{n}(K,J)$ be the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are elements of an ideal $J$ of $K.$ We describe all Jordan derivations of the matrix ring $R_{n}(K,J)$ in this paper. The main result states that every Jordan derivation $Δ$ of $R_{n}(K,J)$ is of the form $Δ=D+Ω$ where $D$ is a derivation of $R_{n}(K,J)$ and $Ω$ is an extremal Jordan derivation of $R_{n}(K,J).$
This paper has been withdrawn by the author due to that the main results and approaches are closedly parallel to the ones in Lie algebra case.
New solutions of twist equations for universal enveloping algebras U(A_{n-1}) are found. They can be presented as products of full chains F_c of extended Jordanian twists, Abelian factors (rotations) F^R and sets of quasi-Jordanian twists F^J. The latter are the generalizations of Jordanian twists (with 2-dimensional Borel carrier b^2) for special deformed extensions of the Hopf algebra U(b^2). The carrier subalgebra g_P for the composition of these three twists is a nonminimal parabolic subalgebra in sl(n). The parabolic twisting elements F_P are obtained in the explicit form. The details of …
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 …
While studying some properties of linear operators in a Euclidean Jordan algebra, Gowda, Sznajder and Tao have introduced generalized lattice operations based on the projection onto the cone of squares. In two recent papers of the authors of the present paper it has been shown that these lattice-like operators and their generalizations are important tools in establishing the isotonicity of the metric projection onto some closed convex sets. The results of this kind are motivated by metods for proving the existence of solutions of variational inequalities and methods for finding these solutions…
Given a Jordan algebra $A$ and a vector space $V$, we describe and classify all Jordan algebras containing $A$ as a subalgebra of codimension ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \, A)$. Any such algebra is isomorphic to a newly introduced object called \emph{unified product} $A \, \natural \, V$. The crossed/twisted product of two Jordan algebras are introduced as special cases of the unified product and the role of the subsequent problem corresponding to each such product is discussed. The non-abelian cohomology ${\rm H}^2_{\rm nab} \…
Axial identities
本文引入轴心代数中的轴心恒等式，以研究固态子代数的性质，并给出了一个既非约当代数也非Matsuo代数同态像的轴心代数例子。
We generalize Baranov and Shlaka's results about bar-minimal Jordan-Lie and regular inner ideals of finite dimensional associative algebras. Let A be a finite dimensional 1-perfect associative algebras A over an algebraically closed field \mathbb{F} of characteristic p\ge0 and let A' be a subalgebra of A. We prove that for any bar-minimal Jordan-Lie inner ideal B' a A', there is a bar-minimal Jordan-Lie inner ideal of A that contains B' and if B' is regular, then is a regular inner ideal of A that contains B'. We also prove that for any strict orthogonal pair (e',f') in A', there is a strict o…
Multiplicative and Jordan multiplicative maps on structural matrix algebras
本文研究了结构矩阵代数上的乘法和乔丹乘法映射，证明在特定条件下这些映射自动为加法，并完全描述了其形式。
In this paper we show that each non-zero ideal of a twisted generalized Weyl algebra (TGWA) $A$ intersects the centralizer of the distinguished subalgebra $R$ in $A$ non-trivially. We also provide a necessary and sufficient condition for the centralizer of $R$ in $A$ to be commutative, and give examples of TGWAs associated to symmetric Cartan matrices satisfying this condition. By imposing a certain finiteness condition on $R$ (weaker than Noetherianity) we are able to make an Ore localization which turns out to be useful when investigating simplicity of the TGWA. Under this mild assumption we…
Jordan homomorphisms and T-ideals
本文研究了域上结合代数间的Jordan同态，证明了幺环且等于换位子理想的代数上Jordan满射同态可分解为同态与反同态之和，并刻画了可延拓为结合同态的T-理想条件。
In this paper, we study the class of Jordan dialgebras. We develop an approach for reducing problems on dialgebras to the case of ordinary algebras. It is shown that straightforward generalizations of the classical Cohn's, Shirshov's, and Macdonald's Theorems do not hold for dialgebras. However, we prove dialgebraic analogues of these statements. Also, we study polylinear special identities which hold in all special Jordan algebras and do not hold in all Jordan algebras. We find a natural correspondence between special identities for ordinary algebras and dialgebras.
Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity.
I describe the lattice of subalgebras of a one-generator Leibniz algebra. Using this, I show that, apart from one special case, a lattice isomorphism between Leibniz algebras L, L' maps the Leibniz kernel of L to that of L'.
Axial algebras of Jordan type $η$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-η)$, where $η\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $η$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $η\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $η$ are factors of the so-called Matsuo algebras corresp…
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 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…
We study the self-dual Hopf algebra $\h\_{\SP}$ of special posets introduced by Malvenuto and Reutenauer and the Hopf algebra morphism from $\h\_{\SP}$ to to the Hopf algebra of free quasi-symmetric functions $\FQSym$ given by linear extensions. In particular, we construct two Hopf subalgebras both isomorphic to $\FQSym$; the first one is based on plane posets, the second one on heap-ordered forests. An explicit isomorphism between these two Hopf subalgebras is also defined with the help of two transformations on special posets. The restriction of the Hopf pairing of $\h\_{\SP}$ to these Hopf …
This paper is a continued investigation of the structure of Lie algebras in relation to their chief factors, using concepts that are analogous to corresponding ones in group theory. The first section investigates the structure of Lie algebras with a core-free maximal subalgebra. The results obtained are then used in section two to consider the relationship of two chief factors of L being L-connected, a weaker equivalence relation on the set of chief factors than that of being isomorphic as L-modules. A strengthened form of the Jordan-Holder Theorem in which Frattini chief factors correspond i…
From bosonized $\mathfrak{osp}(1|2)$ to liftings of the super Jordan plane
本文通过玻色化与Ore局部化，将超约当平面的提升统一为$\mathfrak{osp}(1|2)$的结构，并分类其有限维模、理想及经典商环。
A polynomial criterion for Jordan elements in a free associative algebra
本文为自由结合代数中的Jordan元素提出一个多项式判据，通过构造$U_n$和$q_n$，将元素属于Jordan子代数等价于单一方程$a\\,q_n(U_n)=0$，并给出显式投影及低度维数。
Jordan algebras over icosahedral cut-and-project quasicrystals
本文研究由二十面体准晶切割投影模型集产生的非周期Jordan代数，证明其生成元与准晶元素一一对应，并揭示接受窗口的H2、H3、H4非晶体学对称性会传递给所得代数，同时给出Fibonacci链、Penrose镶嵌和Elser-Sloane准晶等具体例子。
A symmetric biderivation structure on polynomial algebras and a class of modules over the special Jordan algebra $H_n(K)$ of symmetric matrices
本文研究多项式代数上由偏导数定义的双导子结构，刻画其二次齐次部分与对称矩阵特殊 Jordan 代数的同构、各齐次空间的双模性质、权空间分解、自同构群及特征零下的单性。
Jordan embeddings and linear rank preservers of structural matrix algebras
本文研究结构矩阵代数的 Jordan 嵌入与线性秩保持映射，证明任意 SMA 中可对角化交换族可内在同时对角化，刻画 SMA 间的 Jordan 嵌入及自同构，并给出线性秩保持映射的完整形式描述。
We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields, and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras, and provide a series of embeddings between quivers, incidence algebras …
Particle models from special Jordan backgrounds and spectral triples
该研究基于若尔当坐标代数定义谱三元组与代数背景，提出涨落狄拉克算子的规范不变玻色构型空间并完成计算，指出其不等同于常规结合非交换几何，且若尔当情形下规范场恒为幺模，解决了非交换几何的长期难题。
Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities $a(bc) = a(cb)$, $(ba)a^2 = (ba^2)a$, and $(b,a^2,c) = 2(b,a,c)a$. These identities are satisfied by the product $ab = a \dashv b + b \vdash a$ in an associative dialgebra. We use computer algebra to show that every identity for this product in degree $\le 7$ is a consequence of the three identities in degree $\le 4$, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.
In this paper, we extend the theory of special local Moufang sets. We construct a local Moufang set from every local Jordan pair, and we show that every local Moufang set satisfying certain (natural) conditions gives rise to a local Jordan pair. We also explore the connections between these two constructions.
We observe several facts and make conjectures about commutative algebras satisfying the Jacobi identity. The central question is which of those algebras admit a faithful representation (i.e., in Lie parlance, satisfy the Ado theorem, or, in Jordan parlance, are special).
In this paper, we mainly study a class of biderivations and triple homomorphisms on perfect Jordan algebras. Let $J$ be a Jordan algebra and $δ:J \times J \rightarrow J$ a symmetric biderivation satisfying $δ(w , u \circ v) = w \cdot δ(u , v), \forall u,v,w \in J$. If $J$ is perfect and satisfies $Z(J) = \{0\}$, then $δ$ is of the form $δ(x , y) = γ(x \circ y)$ for all $x , y \in J$, where $γ\in Cent(J)$ satisfying $z \cdot γ(x \circ y) = x \cdot γ(y \circ z) + y \cdot γ(x \circ z), \forall x , y , z \in J$. This is the special case of our main theorem which concerns biderivations having their…
Axial algebras are a class of commutative non-associative algebras generated by idempotents, called axes, with adjoint action semi-simple and satisfying a prescribed fusion law. Axial algebras were introduced by Hall, Rehren and Shpectorov \cite{hrs,hrs1} as a broad generalisation of Majorana algebras of Ivanov, whose axioms were derived from the properties of the Griess algebra for the Monster sporadic simple group. The class of axial algebras of Monster type includes Majorana algebras for the Monster and many other sporadic simple groups, Jordan algebras for classical and some exceptional si…
Special identities for the pre-Jordan product in the free dendriform algebra
本文利用对称群表示论和计算线性代数研究自由树形代数中预Jordan积的恒等式，证明次数≤7的恒等式均由4次恒等式推出，并发现次数为8的新恒等式，且与结合dialgebra中Jordan双积的特殊恒等式具有S8-模同构关系。
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…
In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H.
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。
We consider a Borel subalgebra $\fg$ of the general linear algebra and its subalgebra $\BB$ which is a Borel subalgebra of the special linear algebra, over arbitrary field. Let $\cL\in\{\fg, \BB\}$. We establish here explicit realizations of the center $Z(\cL)$ and semi-center $Sz(\cL)$ of the enveloping algebra, the Poisson center $S(\cL)^{\cL}$ and Poisson semi-center $S(\cL)^{\cL}_{\si}$ of the symmetric algebra. We describe their structure as commutative rings and establish isomorphisms $Z(\cL)\cong S(\cL)^{\cL}$, $Sz(\cL)\cong S(\cL)^{\cL}_{\si}$
We consider a Weitzenböck derivation $Δ$ acting on a polynomial ring $R=K[ξ_1,ξ_2,...,ξ_m]$ over a field $K$ of characteristic 0. The $K$-algebra $R^Δ= \{h \in R \mid Δ(h) = 0\}$ is called the algebra of constants. Nowicki considered the case where the Jordan matrix for $Δ$ acting on $R_1$, the degree 1 component of $R$, has only Jordan blocks of size 2. He conjectured (\cite{N}) that a certain set generates $R^Δ$ in that case. Recently Koury (\cite{Kh}), Drensky and Makar-Limanov (\cite{DM}) and Kuroda (\cite{K}) have given proofs of Nowicki's conjecture. Here we consider the case where the J…
An Exceptional 7-dimensional Real Algebra: Octonions, $G_2$, and the Fano Plane
研究一种7维非结合代数，通过八元数叉积将1882年Hüseyin Tevfik Pasha引入的双线性乘法从三维扩展到七维，该代数具有单位元、简单且非结合，其自同构群为U(3)。
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes or quaternions. The remaining, exceptional Jordan algebra can be described by 3x3 Hermitian matrices over the octonions. We first review properties of the octonions and the exceptional Jordan algebra, including our previous work on the octonionic Jordan eigenvalue problem. We then examine a particular real, noncompact form of the Lie group E6, which preserve…
We study graded right coideal subalgebras of Nichols algebras of semisimple Yetter-Drinfeld modules. Assuming that the Yetter-Drinfeld module admits all reflections and the Nichols algebra is decomposable, we construct an injective order preserving and order reflecting map between morphisms of the Weyl groupoid and graded right coideal subalgebras of the Nichols algebra. Here morphisms are ordered with respect to right Duflo order and right coideal subalgebras are ordered with respect to inclusion. If the Weyl groupoid is finite, then we prove that the Nichols algebra is decomposable and the a…
Symmetrization of Jordan dialgebras
该研究针对特殊Jordan对代数乘积对称化的多项式恒等式问题，用计算线性代数与对称群表示论构造性方法，明确了5次及以下恒等式源于交换性，找到6次非交换性推论的恒等式生成元，并证明7次存在新恒等式。
On $3$-generated axial algebras of Jordan type $\frac{1}{2}$
本文针对二次闭域上的3-生成Jordan型1/2轴代数，在已有带4参数的泛代数构造基础上，完整刻画了所有这类半单代数，推进了非结合轴代数的分类研究。
Jordan algebras and weight modules
本文研究 Jordan 代数相关的 ${\mathfrak{sl}}_2(J)$ 模，引入 Weyl 模并证明自由 Jordan 代数情形下存在，猜想其构成最高权范畴，以解决上同调消失问题。
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