The abstract defines G2-structures and connects them to octonion algebras.
problem Classifying G2-structures and understanding their geometric properties. method Established an isomorphism between G2-structures and octonion algebras over C∞(M). result The classification of G2-structures agrees with a parametrisation of octonion algebras with isometric norm. Constructs projective plane over octonions, proving no higher real division algebras.
problem Existence of higher-dimensional real division algebras.
method Using Adams' solution of the Hopf invariant 1 problem, constructs projective plane over octonions.
result No higher-dimensional real division algebras exist.
Research examines octonionic slice regular functions and their automorphisms and invariants.
problem Analyzing slice regular functions in the octonionic algebra.
method Investigates automorphisms and invariants of octonionic slice regular functions.
result Characterizes the automorphisms and invariants of octonionic slice regular functions.
The study finds Lie algebra formulae and classifies polar actions on a hyperbolic plane.
problem Finding Lie algebra formulae and classifying actions on hyperbolic planes.
method Using octonions and triality, explicit Lie brackets were found for Lie algebras of isometry groups.
result Explicit formulae for Lie brackets of f4 and f4∗ Lie algebras. The paper explores spinors and polyforms using quaternions and octonions.
problem Understanding spinors and polyforms in Clifford algebras.
method Generalizes Pauli matrices to quaternions and octonions, and relates these to spinor models.
result Explicitly describes Weyl spinors of Spin(4,4) related to quaternions and octonions.
Formula derived for Lie algebra E8's bracket.
problem Calculating the bracket of the exceptional Lie algebra E8.
method Based on triality and oct-octonions, following Barton-Sudbery description.
result Explicit formula for E8's bracket.
We use the octonion algebra to construct singular solutions of Hessian fully nonlinear uniformly elliptic equations in 21 or more dimensions. The regularity of these solutions is the least possible one. The same is proven for Isaacs equtions.
Special orthogonal representations from octonions have geometric properties linked to binary cubics.
problem Understanding geometric properties of special orthogonal representations from octonions.
method Using octonions and their derivations, spinors, and covariants to show geometric properties.
result Covariants and Mathews identities of these representations are related to the Fano plane and (Z2)3. Introduces Plücker coordinates for a complex projective octonion plane, solving an overdetermined system of relations.
problem Understanding the complex projective octonion plane and its quotient space EIII.
method Introduces Plücker coordinates and uses Clifford algebra to solve the overdetermined system of relations.
result Shows that EIII can be decomposed into F4-orbits and provides detailed analysis near the subvariety X∞.
The purpose of this paper is to provide an octonionic description of the Lie group SL(2,O). The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from h2(O) onto itself. An interesting characterization is giv…
New formula found for a unique invariant 8-form on Riemannian manifolds with Spin(9) structure.
problem Finding a new explicit algebraic formula for a unique invariant 8-form.
method Generalizing the standard Kähler 2-form expression, constructing the invariant 8-form from octonion-valued coordinate 1-forms.
result A new explicit algebraic formula for the Spin(9)-invariant 8-form. Equations found for a special geometric space.
problem Defining equations for a specific geometric space.
method Study of torus fixed points to compute Poincaré polynomial.
result Poincaré polynomial computed for the compactification.
In the quatenions ($\H$, $\H'$, $\H^{C}$) and octonions ($\gC$, $\gC^\prime$, $\gC^C$), we show some results on the conjugacy of two pure imaginary non-zero elements with same norm.
This is an expository paper. Its purpose is to explain the linear algebra that underlies Donaldson-Thomas theory and the geometry of Riemannian manifolds with holonomy in G2 and Spin(7).
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…
Notes on Lie algebra and group G2 for a workshop.
problem Understanding G2 Lie algebra and group. method Algebraic approach, focusing on Lie algebra level.
result Comprehensive survey of G2 including its Lie algebra and group properties. New octonionic Kähler metrics solve an octonionic Calabi-Yau theorem.
problem Finding metrics on 16D manifolds.
method Introduced octonionic Kähler metrics and solved an octonionic Monge-Ampère equation.
result Solved an octonionic Calabi-Yau theorem.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
New hyperbolic manifolds discovered that fiber algebraically up to dimension 8.
problem Finding hyperbolic manifolds that fiber algebraically in all dimensions 5 to 8.
method Assigning colors and states to right-angled hyperbolic polytopes and applying arguments from Jankiewicz et al.
result First examples of hyperbolic manifolds with finitely presented but not of finite type fundamental groups.
We use a G2-structure on a 7-dimensional Riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non-associative product. We then define a metric-compatible octonion covariant derivative on this bundle that is compatible with the octonion product. The torsion of the G2-structure is then sh…
In this sequel, employing more commutative algebra than that explored in \cite{CCJ}, we show that an isoparametric hypersurface with four principal curvatures and multiplicities (3,4) in S15 is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-Münzner \cite{FKM}, referred to collectively as of OT-…
Study on octonionic Nahm's equations and their moduli space properties.
problem Properties of octonionic Nahm's equations and their moduli space.
method Analyzing basic properties, constructing solutions, introducing symmetry, proving theorems.
result Moduli space of smooth solutions to octonionic Nahm's equations over [0,1] is a star-shaped smooth manifold.
Study smooth loops and loop bundles, relating to G2-structures.
problem Properties of smooth loops and their applications.
method Analyze smooth loops, introduce loop bundles, define torsion and curvature.
result Showed how loop bundles relate to G2-structures. Study subelliptic heat kernel on octonionic anti-de Sitter space.
problem Heat kernel of octonionic anti-de Sitter space.
method Lift Laplacian of octonionic hyperbolic space and use sub-Laplacian.
result Two integral representations for subelliptic heat kernel.
New classification of 16D planes with specific automorphism groups.
problem Classifying 16-dimensional projective planes with certain automorphism groups.
method Detailed analysis and classification of 16-dimensional planes with specific automorphism properties.
result Classification of 16-dimensional planes with a group of dimension at least 35, excluding planes fixed by exactly one flag.
In this article, the clarification to Note 4 (arXiv:1202.0941) for n=8 is considered. In this connection, answers to the following questions are given. 1. How to classify the metric hypercomplex orthogonal group alternative-elastic algebras for n=8? 2. How to associate the metric hypercomplex orthogonal group alternati…
New mechanics on non-associative octonions discovered.
problem Discrete mechanics on non-associative groups.
method Generalized Lagrangian and Hamiltonian mechanics to non-associative objects.
result Discrete mechanics on unitary octonions achieved.
Study of SL(2) over octonions using twistor geometry.
problem No invertible 2x2 matrices over octonions, use Spin(9,1) x SL(2,R) orbit.
method Twistor geometry in eight dimensions.
result Interpretation of open orbit in 32D representation space.
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In…
Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz superstring. In both cases, supersymmetry relies on the vanishing of a certain tr…
James's octonionic Stiefel spaces questions answered partially.
problem Two fundamental questions about octonionic Stiefel spaces.
method Partial answers to James's questions about octonionic Stiefel spaces.
result Partial answers to James's questions about octonionic Stiefel spaces.
Construct Clifford systems on Euclidean spaces and manifolds.
problem No specific problem stated; general Clifford systems construction.
method Inductive construction and adaptation to manifolds.
result Developments in octonionic geometry.
Introduction to G2 geometry for beginners.
problem Understanding G2 geometry and its special algebraic structure.
method Informal introduction with emphasis on octonions and linear algebra.
result Explains the special linear algebraic structure in 7 dimensions.
We prove that Riemannian Spin(7) holonomy manifolds carry octonionic-Kähler structure.
Reviews interactions between Spin(9) and octonionic geometries.
problem Understanding the role of Spin(9) in octonionic geometry.
method Analyzes canonical 8-forms, vector fields, Hopf fibrations, and manifolds.
result Discovers new insights into the geometry of octonionic Hopf fibrations.
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
problem Characterizing minimal submanifolds in locally symmetric spaces.
method Analyzing higher expansion properties and volume constraints.
result Codimension two minimal submanifolds have at least linear volume in the ambient space.
Abstract: Investigates octonion product deformations and related geometries.
problem Exploring geometries and deformations from the 7-sphere S7. method Analyzing the spontaneous compactification M4imesS7 and solutions of Lagrangian equations. result Obtains a family of geometries including those with torsion and G2-structures. Sedenions have geometric submanifolds with special properties.
problem Characterizing zero divisors in sedenion algebra.
method Analyzing zero divisors as submanifolds and proving isometries.
result Zero divisors form submanifolds isometric to Lie groups and Stiefel manifolds.
Study subelliptic heat kernel on lifted sphere from octonionic projective space.
problem Analyzing sub-Laplacian on lifted sphere from octonionic projective space.
method Explicit formulas for heat kernel and Green function derived.
result Explicit formulas for heat kernel and Green function.
The dimensions of the spaces of k-homogeneous Spin(9)-invariant valuations on the octonionic plane are computed using results from the theory of differential forms on contact manifolds as well as octonionic geometry and representation theory. Moreover, a valuation on Riemannian manifolds of particular inte…
The aim of this paper is to offer an overview of the most important applications of Jordan structures inside mathematics and also to physics, up-dated references being included. For a more detailed treatment of this topic see - especially - the recent book Iordanescu [364w], where sugestions for further developments ar…
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
New G2 symmetry found in 8D distribution with 6D square.
problem Discovering new G2 symmetry in geometric distributions. method Analyzing rank 3 distribution on 8D manifold with growth vector (3,6,8).
result Maximally symmetric rank 3 distribution with 6D square.
This paper is devoted to the specific class of pseudoconformal mappings of quaternion and octonion variables. Normal families of functions are defined and investigated. Four criteria of a family being normal are proven. Then groups of pseudoconformal diffeomorphisms of quaternion and octonion manifolds are investigated…
Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We exten…
Study of a G2-equivariant octonionic operator and its right spectrum.
problem Understanding the spectrum of a G2-equivariant octonionic operator. method Computed the ordinary real spectrum and analyzed the octonionic right-eigenvalue problem using G2-decomposition and residual symmetry analysis. result Explicit spectral loci (quartic curve and circle) in each complex slice of the octonionic space.
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations of octonionic projective plane.
problem Constructing and classifying triangulations of the octonionic projective plane.
method Combinatorial construction and analysis of symmetry groups.
result Found 634 vertex-transitive and over 10^103 non-vertex-transitive 27-vertex triangulations.