arXiv research

The paper investigates gradings of complex simple Lie algebras, focusing on $|3|$-gradings and their algebraic structures.

The paper finds formulas for flat models of certain Lie algebras.

Formula derived for Lie algebra E8's bracket.

A classification is given of the exceptional $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric spaces $G/K$ by A.Kollross, where $G$ is an exceptional compact Lie group or $S\!pin(8)$, and moreover the structure of $K$ is determined as Lie algebra. In the present article, we give a pair of commuting involutive automorphisms…

A new quantum relation connects exceptional Lie algebras and knots.

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 Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…

We present a geometric construction of the exceptional Lie algebras F4 and E8 starting from the round 8- and 15-spheres, respectively, inspired by the construction of the Killing superalgebra of a supersymmetric supergravity background. (There is no supergravity in the paper.)

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 orbit types in the Lie algebra $\gg_2$ as $$ G_2/G_2, \quad G_2/(U(1) \times U(1)), …

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimensio…

The hidden M-algebra is integrated into a super-Lie group, allowing for compactification of extra dimensions.

We present structural properties of Lie algebras admitting symmetric, invariant and nondegenerate bilinear forms. We show that these properties are not satisfied by nilradicals of parabolic subalgebras of real split forms of complex simple Lie algebras, neither by 2-step nilpotent Lie algebras associated with graphs, w…

Unified framework for exceptional and generalised geometry, and Poisson-Lie duality.

The paper examines alternative definitions of exceptional Lie groups using quaternion numbers.

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-d…

We know that any element $X$ of the exceptional Jordan algebra $\gJ$ is transformed to a diagonal form by the compact exceptional Lie group $F_4$. However, its proof is used the method which is reduced a contradiction. In this paper, we give a direct and constructive proof.

New proof shows Lie algebras are rigid under certain conditions.

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 of the exceptional Lie groups: $G_2$, its definition and different characterizati…

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo $H$-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type $B_n$ with $|2|$-grading do not contain non-Heisenberg pseudo $H$-type Li…

Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.

Machine learning accelerates Lie algebra computations.

Research explores Kähler and semi-para-Kähler structures on specific Lie groups.

Proofs centerless unimodular contact Lie algebras.

The study identifies exceptions to fiber-preserving symmetry in ODEs and systems.

The paper explores alternative definitions of complex Lie groups using real numbers.

We study isometric Lie group actions on the compact exceptional groups E6, E7, E8, F4 and G2 endowed with a biinvariant metric. We classify polar actions on these groups. We determine all isometric actions of cohomogeneity less than three on E6, E7, F4 and all isometric actions of cohomogeneity less than 20 on E8. More…

The notion of (Z/2Z) x (Z/2Z)-symmetric spaces is a generalization of classical symmetric spaces, where the group Z/2Z is replaced by (Z/2Z) x (Z/2Z). In this article, a classification is given of the (Z/2Z) x (Z/2Z)-symmetric spaces G/K where G is an exceptional compact Lie group or Spin(8), complementing recent resul…

The paper extends a geometric model using singular curves.

We give necessary and sufficient conditions of the existence of a left-invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra $\mathfrak{g}$ is a filiform Lie algebra $\mathfrak{n}$. It turns out that such a metric always exists, except for in the two cases, wh…

Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…

We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. On one hand, our approach is based on the algorithms developed in this wo…

Classifies non-integrable distributions with simple infinite-dimensional Lie superalgebras of symmetries.

We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for th…

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-inv…

Completed realizations of automorphisms and subgroups in exceptional Lie group $E_8$.

For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{d…

Special orthogonal representations from octonions have geometric properties linked to binary cubics.

Study on deformations of symmetric spaces using Jordan algebras.

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are constructed and the algebra structures are computed. Finally, the kinematic formula…

Constructing transitive nilpotent Lie algebras from dilations and analyzing their prolongations.

We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group $\mathrm{String}(n)$. A Lie 2-algebra is a categorified version of a Lie algebra where the Jacobi identity holds up to a natural isomorphism called the "Jacobiator". Similarly, a Lie 2-group is a c…

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 structure of stabilizer for each $G$-orbit on $\mathcal{J}^1$.

Pulling back the weight system associated with the exceptional Lie algebra G_2 by a modification of the universal Vassiliev-Kontsevich invariant yields a link invariant; extending it to 3-nets, we derive a recursive algorithm for its evaluation.

Study of exceptional algebroids in relation to type IIB superstrings.

New Virasoro-like structures for circle diffeomorphisms with breaks.

Study on simplicity of Lie skew braces, proving new results for compact cases.

Understanding the exceptional Lie groups as the symmetry groups of simpler objects is a long-standing program in mathematics. Here, we explore one famous realization of the smallest exceptional Lie group, G2. Its Lie algebra acts locally as the symmetries of a ball rolling on a larger ball, but only when the ratio of r…

We give local descriptions of parabolic contact structures and show how their flat models yield explicit PDE having symmetry algebras isomorphic to all complex simple Lie algebras except $\mathfrak{sl}_2$. This yields a remarkably uniform generalization of the Cartan-Engel models from 1893 in the $G_2$ case. We give a …

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a deriv…