相关论文: A generalization of the Kantor-Koecher-Tits constr…
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…
The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra $\langle A, \Delta\rangle$, it is possible to construct a Lie coalgebra $\langle L(A), \Delta_{L}\rangle$. Moreover, any dual algebra…
We present a nonlinear realization of the 5-graded Lie algebra associated to a Kantor triple system. Any simple Lie algebra can be realized in this way, starting from an arbitrary 5-grading. In particular, we get a unified realization of…
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…
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…
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…
We study the relationship between cyclic homology of Jordan superalgebras and second cohomologies of their Tits-Kantor-Koecher Lie superalgebras. In particular, we focus on Jordan superalgebras that are Kantor doubles of bracket algebras.…
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…
Four-dimensional extended: Poincar\'e, AdS-Lorentz and Maxwell algebras, are obtained by expanding an extension of de Sitter or conformal algebra, SO(4,1) or SO(3,2). The procedure can be generalized to obtain a new family of extended…
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…
Kantor pairs, (quadratic) Jordan pairs, and similar structures have been instrumental in the study of $\mathbb{Z}$-graded Lie algebras and algebraic groups. We introduce the notion of an operator Kantor pair, a generalization of Kantor…
Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose…
On the set H_n(K) of symmetric n by n matrices over the field K we can define various binary and ternary products which endow it with the structure of a Jordan algebra or a Lie or Jordan triple system. All these non-associative structures…
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…
Each choice of a K\"ahler class on a compact complex manifold defines an action of the Lie algebra $\slt$ on its total complex cohomology. If a nonempty set of such K\"ahler classes is given, then we prove that the corresponding…
We consider generalization of wellknown construction Kantor Double J({\Gamma}, {,}) (KKM Double, Kantor-King-McCrimmon Double), where basic algebra {\Gamma} is nonunital algebra. We find necessary and sufficient conditions for a generalized…
Two types of higher order Lie $\ell$-ple systems are introduced in this paper. They are defined by brackets with $\ell > 3$ arguments satisfying certain conditions, and generalize the well known Lie triple systems. One of the…
In this paper we describe an approach to generalise minimal representations to the super setting for Lie superalgebras obtained from Jordan superalgebras using the TKK construction. This approach was used successfully to construct a Fock…
The purpose of this study is to extend the concept of a generalized Lie $3-$ algebra, known to the divisional algebra of the octonions $\mathbb{O}$, to split-octonions $\mathbb{SO}$, which is non-divisional. This is achieved through the…
We present the subalgebra structure of sl(3,O), a particular real form of e6 chosen for its relevance to particle physics and its close relation to generalized Lorentz groups. We use an explicit representation of the Lie group SL(3,O) to…