Related papers: On realizations of the Lie groups $ G_{2,\boldmath…

In order to define the complex exceptional Lie groups $ {F_4}^C, {E_6}^C, {E_7}^C, {E_8}^C $ and these compact real forms $ F_4,E_6,E_7,E_8 $, we usually use the Cayley algebra $ \mathfrak{C} $. In the present article, we consider replacing…

There exist six Lie groups of type $ E_6 $, and to be specific, ${E_6}^C , E_6, E_{6(6)}, E_{6(-2)}, E_{6(-14)}, E_{6(-26)}$. In order to define these groups, we use usually the Cayley algebra $ \mathfrak{C} $ and the split Cayley algebra $…

These notes have been prepared for the Workshop on "(Non)-existence of complex structures on $\mathbb{S}^6$", to be celebrated in Marburg in March, 2017. The material is not intended to be original. It contains a survey about the smallest…

In physics, Lie groups represent the algebraic structure that describes symmetry transformations of a given system. Then, the descending Lie algebra of those groups are necessarily real. In most cases, the complexification of those Lie…

A new structure, based on joining copies of a group by means of a \emph{twist}, has recently been considered to describe the brackets of the two exceptional real Lie algebras of type $G_2$ in a highly symmetric way. In this work we show…

In this article we provide a detailed description of a technique to obtain a simple parametrization for different exceptional Lie groups, such as G2, F4 and E6, based on their fibration structure. For the compact case, we construct a…

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the…

A new highly symmetrical model of the compact Lie algebra $\mathfrak{g}^c_2$ is provided as a twisted ring group for the group $\mathbb{Z}_2^3$ and the ring $\mathbb{R}\oplus\mathbb{R}$. The model is self-contained and can be used without…

We introduce a remarkable subset "the stem" of the set of positive roots of a reduced root system. The stem determines several interesting decompositions of the corresponding reductive Lie algebra. It gives also a nice simple three…

A linear algebraic group G is over a field K is called a Cayley K-group if it admits a Cayley map, i.e., a G-equivariant K-birational isomorphism between the group variety G and its Lie algebra. We classify real reductive algebraic groups…

For simply connected compact exceptional Lie groups $G = F_4, E_6$ and $E_7$, we consider two involutions $\sigma, \gamma$ and determine the group structure of subgroups $G^{\sigma,\gamma}$ of $G$ which are the intersection $G^\sigma \cap…

This paper is a continuation of [5]. Using the root categories, we define the compact real forms of the complex semisimple Lie algebras, and maximal compact subgroups of the Chevalley groups over $\mathbb{C}$. In [7], Lusztig used the…

We give a construction of the compact real form of the Lie algebra of type $E_6$, using the finite irreducible subgroup of shape $3^{3+3}:\mathrm{SL}_3(3)$, which is isomorphic to a maximal subgroup of the orthogonal group $\Omega_7(3)$. In…

We fill in the "hole" in the exceptional series of Lie algebras that was observed by Cvitanovic, Deligne, Cohen and deMan. More precisely, we show that the intermediate Lie algebra between $E_7$ and $E_8$ satisfies some of the decomposition…

The five exceptional simple Lie algebras over the complex number are included one within the other as $G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8$. The biggest one, $E_8$, is in many ways the most mysterious. This article surveys…

Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for…

We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of…

We describe simply connected compact exceptional simple Lie groups in very elementary way. We first construct all simply connected compact exceptional Lie groups G concretely. Next, we find all involutive automorphisms of G, and determine…

A weakly complete vector space over $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ is isomorphic to $\mathbb{K}^X$ for some set $X$ algebraically and topologically. The significance of this type of topological vector spaces is…

Let $G$ be a connected reductive algebraic group over an algebraically closed field $\mathbf{k}$, and let Lie$(G)$ be its associated Lie algebra. In his series of papers on unipotent elements in small characteristic, Lusztig defined a…