Alberto Elduque, Noriaki Kamiya, Susumu Okubo
A noncommutative Jordan algebra of a specific type is attached to any (-1,-1)-balanced Freudenthal Kantor triple system, in such a way that the triple product in this system is determined by the binary product in the algebra. Over fields of characteristic zero, the simple noncommutative Jordan algebras of this type are classified.
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Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3 are considered. The close connections of these algebras with Lie algebras graded over the nonreduced root system BC1, with J-ternary algebras and with Freudenthal-Kantor triple systems are explored.
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.
Left unital Kantor triple systems will be shown to correspond to structurable algebras endowed with an involutive automorphism. A related result is proved for (-1,-1) Freudenthal-Kantor triple systems. Some consequences for the associated 5-graded Lie algebras and superalgebras are deduced too. In particular, left unital (-1,-1) Freudenthal-Kantor triple systems are shown to be intimately related to Lie superalgebras graded over the root system of type B(0,1).
Homogeneous Freudenthal algebras and the first Tits construction
本文研究了弗雷德霍尔代数在何种情况下是同构的，并通过第一蒂茨构造与离散赋值下的弗雷德霍尔除法代数联系起来，探讨了其同源性条件和局部-整体原理。
We determine explicit orbit representatives of reducible Jordan algebras and of their corresponding Freudenthal triple systems. This work has direct application to the classification of extremal black hole solutions of N = 2, 4 locally supersymmetric theories of gravity coupled to an arbitrary number of Abelian vector multiplets in D = 4, 5 space-time dimensions.
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.
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.
We consider generalization of wellknown construction Kantor Double J(Γ, {,}) (KKM Double, Kantor-King-McCrimmon Double), where basic algebra Γ is nonunital algebra. We find necessary and sufficient conditions for a generalized Kantor double to be Jordan. We also describe $δ$-superderivations of a generalized Kantor double whose even part is prime.
We study a noncommutative generalization of Jordan algebras called Jordan dialgebras. These are algebras that satisfy the identities $[x_1 x_2]x_3= 0$, $(x_1^2,x_2,x_3)=2(x_1,x_2,x_1x_3)$, $x_1(x_1^2 x_2)=x_1^2(x_1 x_2)$; they are related with Jordan algebras in the same way as Leibniz algebras are related to Lie algebras. We present an analogue of the Tits---Kantor---Koecher construction for Jordan dialgebras that provides an embedding of such an algebra into Leibniz algebra.
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。
We find a class of hermitian generalized Jordan triple systems (HGJTSs) and hermitian $(ε, δ)$-Freudenthal-Kantor triple systems (HFKTSs). We apply one of the most simple HGJTSs which we find to a field theory, and obtain a typical u(N) Chern-Simons gauge theory with a fundamental matter.
We show that the three-qubit entanglement classes: (0) Null, (1) Separable A-B-C, (2a) Biseparable A-BC, (2b) Biseparable B-CA, (2c) Biseparable C-AB, (3) W and (4) GHZ correspond respectively to ranks 0, 1, 2a, 2b, 2c, 3 and 4 of a Freudenthal triple system defined over the Jordan algebra C+C+C. We also compute the corresponding SLOCC orbits.
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。
Operator Kantor Pairs
本文引入了算子Kantor对的概念，将其推广至任意交换含单位元环，构造了相关的5-分次Lie代数，并在特定环上建立了其与Kantor对的一一对应关系。
We give a classification up to equivalence of the fine group gradings by abelian groups on the Jordan pairs and triple systems of types bi-Cayley and Albert, under the assumption that the base field is algebraically closed of characteristic different from $2$. The associated Weyl groups are computed. We also determine, for each fine grading on the bi-Cayley and Albert pairs, the induced grading on the exceptional simple Lie algebra given by the Tits-Kantor-Koecher construction.
Automorphism group schemes of Kantor pairs of associative central simple structurable algebras
研究结合中心单可构造代数的 Kantor 对及相关三元系在特定条件下的自同构群概型，针对三种情形中的两种，将自同构群概型的中心积分解与 Kantor 对的张量积分解相联系。
We consider an integral version of the Freudenthal construction relating Jordan algebras and exceptional algebraic groups. We show how this construction is related to higher composition laws of M.Bhargava in number theory. We propose an algorithmic approach to studying orbit spaces of groups underlying higher composition laws. Using this method we discover two new examples of spaces sharing similar properties, and indicate several more examples of spaces where such composition laws may be introduced.
Jordan 3-graded Lie algebras with polynomial identities
本文借助 Tits-Kantor-Koecher 构造将 PI 条件转化为相关 Jordan 对的性质，证明强素情形下的 Posner-Rowen 型定理，并利用 Kostrikin 根刻画一般 PI Jordan 3-分次 Lie 代数。
Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterpart of the isotropy irreducible homogeneous spaces. These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are cla…
Leibniz algebras and their connection to Jordan pair disystems
本文研究莱布尼茨代数与约旦对双系的关联，给出约旦对双系的Tits-Kantor-Koecher构造，引入其同伦与莱布尼茨代数的阿贝尔内理想，扩展了约旦dialgebras的相关构造。
Solutions to the exercises from the book "Albert algebras over commutative rings"
本文提供了《在交换环上Albert代数》一书习题的解答，包括对Albert代数性质的证明及相关补充内容。
The Kantor-Koecher-Tits construction associates a Lie algebra to any Jordan algebra. We generalize this construction to include also extensions of the associated Lie algebra. In particular, the conformal realization of so(p+1,q+1) generalizes to so(p+n,q+n), for arbitrary n, with a linearly realized subalgebra so(p,q). We also show that the construction applied to 3x3 matrices over the division algebras R, C, H, O gives rise to the exceptional Lie algebras f4, e6, e7, e8, as well as to their affine, hyperbolic and further extensions.
In this note we connect finite presentability of a Jordan algebra to finite presentability of its Tits-Kantor-Koecher algebra. Through this we complete our discussion of finite presentability of universal central extensions of ${\mathfrak{sl}_n(A)}$, $A$ a $k$-algebra, initiated in our previous paper, and answer a question raised by Shestakov-Zelmanov in the positive.
On the conjecture of Kashuba and Mathieu about free Jordan algebras
本文通过总结自由Jordan代数的新计算数据，解释了为何在大量正面证据下，Kashuba和Mathieu关于Lie代数同调消没的猜想并不成立。
We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii. Both classifications are consequences of a new description of Jordan algebras …
In 1990 Kantor defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of $W(2).$ Also similar problems are solved for the algebra $W_2$ of all commutative algebras on the 2-dimensional vector space and for the algebra $S_2$ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.
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…
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。
The notion of conservative algebras appeared in a paper by Kantor in 1972. Later, he defined the conservative algebra $W(n)$ of all algebras (i.e. bilinear maps) on the $n$-dimensional vector space. If $n>1$, then the algebra $W(n)$ does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). It looks like $W(n)$ in the theory of conservative algebras plays a similar role to the role of $\mathfrak{gl}_n$ in the theory of Lie algebras. Namely, an arbitrary conservative algebra can be obtained from a universal algebra $W(n)$ for some $n \in \mathbb…
Conservative algebras of $2$-dimensional algebras, IV
本文研究二维向量空间上所有代数构成的保守代数W(2)及其主子代数，延续Kantor提出的保守代数理论，揭示W(n)在保守代数中类似gl_n在Lie代数中的核心作用。
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