Related papers: Freudenthal triple systems by root system methods

The construction of Freudenthal's Magic Square, which contains the exceptional simple Lie algebras, in terms of symmetric composition algebras is further developed here. The para-Hurwitz algebras, which form a subclass of the symmetric…

A $G$-grading on a complex semisimple Lie algebra $L$, where $G$ is a finite abelian group, is called quasi-good if each homogeneous component is 1-dimensional and 0 is not in the support of the grading. Analogous to classical root systems,…

Let $G$ be a connected simple Lie group with finite centre, $K$ be a maximal compact subgroup of $G,$ and rank$(G)=$ rank$(K).$ Let $\frak{g}_0=$Lie$(G), \frak{k}_0=$Lie$(K) \subset \frak{g}_0, \frak{t}_0$ be a maximal abelian subalgebra of…

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…

Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that…

Quaternions, split quaternions, and hybrid numbers are very well-known number systems. These number systems are used to make geometry in Euclidean and Lorentz spaces. These number systems can be obtained with the help of a quadratic form.…

This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the classical root systems, $F_4$ and $E_6$. Such sequences are associated with reduced decompositions (words) in affine and…

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the…

In this paper, we introduce a new and direct approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot, \cdot)$ be a non-degenerate bilinear form and $E$ be a set in…

We find an explicit presentation of relative odd unitary Steinberg groups constructed by odd form rings and of relative doubly laced Steinberg groups over commutative rings, i.e. the Steinberg groups associated with the Chevalley group…

We give a definition of quarternion Lie algebra and of the quarternification of a complex Lie algebra. By our definition gl(n,H), sl(n,H), so*(2n) and sp(n) are quarternifications of gl(n,C), sl(n,C), so(n,C) and u(n) respectively. Then we…

We introduce and provide a classification theorem for the class of Heisenberg-Fock Leibniz algebras. This category of algebras is formed by those Leibniz algebras $L$ whose corresponding Lie algebras are Heisenberg algebras $H_n$ and whose…

Let $\mathfrak{h}_3$ be the Heisenberg algebra and let $\mathfrak g$ be the 3-dimensional Lie algebra having $[e_1,e_2]=e_1\,(=-[e_2,e_1])$ as its only non-zero commutation relations. We describe the closure of the orbit of a vector of…

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations,…

A concise study of ternary and cubic algebras with $Z_3$ grading is presented. We discuss some underlying ideas leading to the conclusion that the discrete symmetry group of permutations of three objects, $S_3$, and its abelian subgroup…

The explicit description of the Frobenius structure for the elliptic root system of type $D_4^{(1,1)}$ in terms of the characters of an affine Lie algebra of type $D_4^{(1)}$ is given.

Freudenthal algebras over a field are basically the same as Jordan algebras of degree $3$ remaining simple under all base field extensions. These algebras are intimately linked, via their automorphism groups and structure groups, to simple…

The fractional stable set polytope ${\rm FRAC}(G)$ of a simple graph $G$ with $d$ vertices is a rational polytope that is the set of nonnegative vectors $(x_1,\ldots,x_d)$ satisfying $x_i+x_j\le 1$ for every edge $(i,j)$ of $G$. In this…

Lagrangian multiforms provide a variational framework for describing integrable hierarchies. This thesis presents two approaches for systematically constructing Lagrangian one-forms, which cover the case of finite-dimensional integrable…

The paper is devoted to classification problem of finite dimensional complex none Lie filiform Leibniz algebras. Actually, the observations show there are two resources to get classification of filiform Leibniz algebras. The first of them…