Related papers: Orbit types of the compact Lie group E_7 in the co…
A Lie group $G$ naturally acts on its Lie algebra $\gg$, called the adjoint action. In this paper, we determine the orbit types of the compact exceptional Lie group $G_2$ in its Lie algebra $\gg_2$. As results, the group $G_2$ has four…
In this paper we investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on a simple Lie algebras of type $E_6$, $F_4$ and $G_2$. The methods for classifying the orbits for…
This paper is a continuation of arXiv:1201.1102. We investigate the orbit closures for the class of representations of simple algebraic groups associated to various gradings on the simple Lie algebra of type $E_7$. The methods for…
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple…
We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type $\mathrm{E}_7$, and we realize groups of type…
Consider a simple complex Lie group $G$ acting diagonally on a triple flag variety $G/P_1\times G/P_2\times G/P_3$, where $P_i$ is parabolic subgroup of $G$. We provide an algorithm for systematically checking when this action has finitely…
By exploiting suitably constrained Zorn matrices, we present a new construction of the algebra of sextonions (over the algebraically closed field $\mathbb{C}$). This allows for an explicit construction, in terms of Jordan pairs, of the…
The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple…
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…
Let $\mathcal{J}^1$ be the real form of complex simple Jordan algebra with the automorphism group $G$ of type $F_{4(-20)}$. Explicitly, we give the orbit decomposition of $\mathcal{J}^1$ under the action of $G$ and determine the Lie group…
For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincar\'{e} polynomials and irreducible components…
Let N be the maximal unipotent subgroup in the simple algebraic group of type {\Phi}. It naturally acts on the space dual to the Lie algebra n of N, and this action is called coadjoint. Such orbits play the key role in the orbit method of…
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…
For certain Lie algebras g, we can use a Z/5Z-grading and define a quartic form and a skew-symmetric bilinear form on the degree 1 component, g_1, thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple…
In this paper we present a construction of the compact form of the exceptional Lie group F4 by exponentiating the corresponding Lie algebra f4. We realize F4 as the automorphisms group of the exceptional Jordan algebra, whose elements are 3…
Flat coordinates for Frobenius manifolds defined on the orbit space of a Coxeter group W are specified through a certain system of generators of W-invariant polynomials. In this note, starting from basic invariants proposed by M.Mehta, we…
In this paper we give an effective method for finding a unique representative of each orbit of the adjoint and coadjoint action of the real affine orthogonal group on its Lie algebra. In both cases there are orbits which have a modulus that…
We show that the Lusternik-Schnirelmann category of a simple, simply connected, compact Lie group G is bounded above by the sum of the relative categories of certain distinguished conjugacy classes in G corresponding to the vertices of the…
Recently, Maurice Chayet and Skip Garibaldi introduced a class of commutative non-associative algebras. In previous work, we gave an explicit description of these algebras for groups of type $G_2,F_4$ and certain forms of $E_6$ in terms of…
We suggest a way to associate to each Lie algebra of type G2, D4, F4, E6, E7, E8 a family of polarized hyperkahler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or…