A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
The abstract defines -structures and connects them to octonion algebras.
Constructs projective plane over octonions, proving no higher real division algebras.
Research examines octonionic slice regular functions and their automorphisms and invariants.
The paper explores spinors and polyforms using quaternions and octonions.
Formula derived for Lie algebra E8's bracket.
Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras and , 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 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.
Introduces Plücker coordinates for a complex projective octonion plane, solving an overdetermined system of relations.
It is well known that there is a unique -invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak -structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the pres…
The purpose of this paper is to provide an octonionic description of the Lie group . The main result states that it can be obtained as a free group generated by invertible and determinant preserving transformations from onto itself. An interesting characterization is giv…
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 and .
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…
New octonionic Kähler metrics solve 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.
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…
Study on octonionic Nahm's equations and their moduli space properties.
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 in is one constructed by Ozeki-Takeuchi \cite[I]{OT} and Ferus-Karcher-Münzner \cite{FKM}, referred to collectively as of OT-…
Study subelliptic heat kernel on octonionic anti-de Sitter space.
Study smooth loops and loop bundles, relating to -structures.
We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative -planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré…
New mechanics on non-associative octonions discovered.
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…
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…
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…
James's octonionic Stiefel spaces questions answered partially.
We prove that Riemannian holonomy manifolds carry octonionic-Kähler structure.
Introduction to G2 geometry for beginners.
Minimal submanifolds in octonionic hyperbolic spaces have large volume.
Abstract: Investigates octonion product deformations and related geometries.
Sedenions have geometric submanifolds with special properties.
The dimensions of the spaces of -homogeneous -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…
We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.
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.
New symmetry found in 8D 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 -equivariant octonionic operator and its right spectrum.
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.
Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the -equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,…
The paper defines and characterizes 2-Ruled hypersurfaces in Minkowski 4-space using octonions.
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The approach is via twistor geometry in eight dimensions.
A novel geometric algebra-based KG embedding framework improves link prediction.
The paper connects polygon spaces with quotient spaces using spin actions and normed division algebras.