On Octonionic Regular Functions and the Szegö Projection on the Octonionic Heisenberg Group

We discuss the octonionic regular functions and the octonionic regular operator on the octonionic Heisenberg group. This is the octonionic version of CR function theory in the theory of several complex variables and regular function theory on the quaternionic Heisenberg group. By identifying the octonionic algebra with \(\mathbb{R }^{8}\) , we can write the octonionic regular operator and the associated Laplacian operator as real \((8\times 8)\) -matrix differential operators. Then we use the group Fourier transform on the octonionic Heisenberg group to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szegö kernel of the orthonormal projection from the space of \(L^{2}\) functions to the space of \(L^{2}\) regular functions on the octonionic Heisenberg group.     

Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

We study in detail the zero set of a slice regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.     

Construction of Octonionic Polynomials

In a previous paper “[On Octonionic Polynomials”, Advances in Applied Clifford Algebras, 17 (2), (2007), 245–258] we discussed generalizations of results on quaternionic polynomials to the octonionic polynomials. In this paper, we continue this generalization searching for methods to construct octonionic polynomials with a prescribed set of zeros.     

The octonionic eigenvalue problem

We discuss the eigenvalue problem for 2×2 and 3×3 octonionic Hermitian matrices. In both cases, we give the general solution for real eigenvalues, and we show there are also solutions with non-real eigenvalues.     

Spin-invariant valuations on the octonionic plane

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 interest is constructed which yields, as a special case, an element of Val₂^Spin (9).     

八元数矩阵的行列式及其性质

赋范的可除代数只有四种:实数R,复数C,四元数日和八元数O.由于八元数关于乘法非交换且非结合,如何对八元数矩阵定义行列式并使其具有较好的运算性质变得非常困难.最近,李兴民和黎丽根据"八元数自共轭矩阵的行列式应为实数"这一数学与物理上的需求,通过选择几个八元数乘积的次序和结合方式,首次给出了八元数行列式的定义.但是,与实数、复数以及四元数的相应的情形比较,如此定义的行列式,其所具备的运算性质较少.本文给出了一种新的八元数行列式的定义,它们具备了尽可能多的运算性质,同时使得"八元数自共轭矩阵的行列式为实数"不证自明.     

Orthonormal basis of the octonionic analytic functions

By confirming a conjecture proposed in Li and Peng (2001) [1], we obtain the orthonormal basis for the octonionic analytic functions.     

About Nonassociativity in Mathematics and Physics

A short review about nonassociative algebraic systems (mainly nonassociative algebras) and their physical applications is presented. We begin with some motivations, then we give a brief historical overview about the formation and development of the concept of hypercomplex number system and about some earlier applications. The main directions discussed are the octonionic, Lie-admissible, and quasigroup approaches. Also, some problems investigated in Tartu, the octonionic approach, Moufang–Mal'tsev symmetry, and associator quantization are discussed. This review does not pretend to be complete as the accent is placed on ideas and not on the techniques, also the references are quite sporadic (there are many authors and results mentioned in the text without references).     

On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.     

On Octonionic Polynomials

We discuss the generalization of results on quaternionic polynomials to the octonionic polynomials. In contrast to the quaternions the octonionic multiplication is non-associative. This fact although introducing some difficulties nevertheless leads to some new results. For instance, the monic and non-monic polynomials do not have, in general, the same set of zeros. Concerning the zeros, it is shown that in the monic and non-monic cases they are not the same, in general, but they belong to the same set of conjugacy classes. Despite these difficulties created by the non-associativity, we obtain equivalent results to the quaternionic case with respect to the number of zeros and the procedure to compute them.     

An improvement of the octonionic Taylor type theorem

We prove that the octonionic polynomials V ■k l 1 ··· l k are independent of the associative orders ■k . This improves the octonionic Taylor type theorem.     

The all-associativity of octonions and its applications

Using an elementary method, we give a new proof of the all-associativity of octonions. As some applications, the known Taylor theorem is improved, and a new definition and new properties of octonionic determinant are also obtained.     

Eigenvalue problem for symmetric 3×3 octonionic matrix

The eigenvalue problem of symmetric 3×3 octonionic matrix has been analyzed. We have especially proved explicitly first that octonionic eigenfunctions have six independent solutions in general with four degeneracy each, and second that for different eigenvalues they satisfy a cubic orthogonality relation under some conditions, which has been previously discovered by Dray and Manogue by computer use. For these, the close relationship between the octonion algebra and a Clifford algebra plays a significant role.     

(Semi-)Riemannian geometry of (para-)octonionic projective planes

We use reduced homogeneous coordinates to construct and study the (semi-)Riemannian geometry of the octonionic (or Cayley) projective plane , the octonionic projective plane of indefinite signature , the para-octonionic (or split octonionic) projective plane and the hyperbolic dual of the octonionic projective plane . We also show that our manifolds are isometric to the (para-)octonionic projective planes defined classically by quotients of Lie groups.     

左O-解析函数紧致奇点的可去性

借助于Adams等人的代数方法,我们证明了多八元数左O-解析函数以及满足一类微分方程的左O-解析函数的紧致奇点的可去性.     

OCTONIONIC HERMITIAN MATRICES WITH NON-REAL EIGENVALUES

We extend previous work on the eigenvalue problem for Hermitian octonionic matrices by discussing the case where the eigenvalues are not real, giving a complete treatment of the 2 × 2 case, and summarizing some preliminary results for the 3 × 3 case.     

Möbius transformation and a version of Schwarz lemma in octonionic analysis

The octonion is a generalization of complex to noncommutative and nonassociative space which has closed relation with exception geometries, wave equation, Yang‐Mills equations, black hole, string theory, and special relativity. In this paper, the Möbius transformation in this manner is first introduced, and some properties are discussed about the transformation in octonionic analysis. Some technique lemmas will be given to solve the problems caused by the weak form of associativity. These versions of Schwarz lemma and Schwarz‐Pick lemma are first studied in octonionic setting which will invoke integral representation formula for harmonic function and Möbius transformations. This will generalize the corresponding results which appear in the classical function theory to nonassociative space and may give new energy for the development of physics.     

Ricci-positive metrics on connected sums of projective spaces

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed Ricci positive metrics on arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct Ricci positive metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.     

Slice regular functions on real alternative algebras

In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C¹.