The octonions are the largest of the four normed division algebras. While
somewhat neglected due to their nonassociativity, they stand at the crossroads
of many interesting fields of mathematics. Here we describe them and their
relation to Clifford algebras and spinors, Bott periodicity, projective and
Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also
touch upon their applications in quantum logic, special relativity and
supersymmetry.Comment: 56 pages LaTeX, 11 Postscript Figures, some small correction

Chang, Woo-Nyoung

Lee, Jae-Hyouk

Lee, Sung Hwan

Lee, Young Jun

Bulletin (new series) of the American Mathematical Society

English

arXiv

summary:We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_{4}$, and the Gosset polytope $4_{21}$ with the symmetry $E_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_{8}$ or $D_{4}$ actions on the $4_{21}$ or the $24$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope $2_{31}$ with the symmetry $E_{7}$, and it is a dual polytope to a Gosset polytope $3_{21}$ with the symmetry $E_{7}$ which is the set of the unit integral octonions with $\operatorname {Re}=1/2$

Czech digital mathematics library

Gosset polytopes in integral octonions

Institute of Mathematics AS CR, v. v. i.

summary:The universe we see gives every sign of being composed of matter. This is considered a major unsolved problem in theoretical physics. Using the mathematical modeling based on the algebra $\bold T:=\bold C\otimes \bold H \otimes \bold O$, an interpretation is developed that suggests that this seeable universe is not the whole universe; there is an unseeable part of the universe composed of antimatter galaxies and stuff, and an extra 6 dimensions of space (also unseeable) linking the matter side to the antimatter---at the very least

Dixon, Geoffrey

Seeable matter; unseeable antimatter

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting elds of mathe-matics. Here we describe them and their relation to Cliord algebras and spinors, Bott period-icity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry. 

John C. Baez

CiteSeerX

The octonions

summary:Let $\mathcal{A}$ be the algebra of quaternions $\mathbb{H}$ or octonions $\mathbb{O}$. In this manuscript an elementary proof is given, based on ideas of Cauchy and D'Alembert, of the fact that an ordinary polynomial $f(t) \in \mathcal{A} [t]$ has a root in $\mathcal{A}$. As a consequence, the Jacobian determinant $\lvert J(f)\rvert $ is always non-negative in $\mathcal{A}$. Moreover, using the idea of the topological degree we show that a regular polynomial $g(t)$ over $\mathcal{A}$ has also a root in $\mathcal{A}$. Finally, utilizing multiplication ($*$) in $\mathcal{A}$, we prove various results on the topological degree of products of maps. In particular, if $S$ is the unit sphere in $\mathcal{A}$ and $h_1, h_2\colon S \to S$ are smooth maps, it is shown that $\deg (h_1 * h_2)=\deg (h_1) + \deg (h_2)$

Sakkalis, Takis

Polynomials and degrees of maps in real normed algebras

summary:We prove that the exceptional complex Lie group ${\mathrm{F}_4}$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of ${\mathrm{Spin}}(9,\mathbb{C})\leq {\mathrm{F}_4}$. Moreover, we identify the stabilizer of the ${\mathrm{F}_4}$-action as a parabolic subgroup ${\mathrm{P}_4}$ (with Levi factor $\mathrm{B_3T_1}$) of the complex Lie group ${\mathrm{F}_4}$. In the real case we obtain an analogous realization of ${\mathrm{F}_4}^{(-20)}/\P$

Pazourek, Karel

Tuček, Vít

Franek, Peter

Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$

summary:Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group $E_6$, and of its subgroups. We are therefore led to a description of $E_6$ in terms of $3\times 3$ octonionic matrices, generalizing previous results in the $2\times 2$ case. Our treatment naturally includes a description of several important subgroups of $E_6$, notably $G_2$, $F_4$, and (the double cover of) $SO(9,1)$. An interpretation of the actions of these groups on the squares of 3-component Cayley spinors is suggested

Dray, Tevian

Manogue, Corinne A.

Octonionic Cayley spinors and $E_6$

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.418.9650

The Octonions

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Czech digital mathematics library

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library

CiteSeerX

Institute of Mathematics AS CR, v. v. i.

Institute of Mathematics AS CR, v. v. i.

Institute of Mathematics AS CR, v. v. i.

Institute of Mathematics AS CR, v. v. i.