Division Algebra Of Octonions Research Articles

Splitting of quadratic Pfister forms over purely inseparable extensions in characteristic 2

We give examples of anisotropic n-fold quadratic Pfister forms (n≥2) over a suitably chosen field F of characteristic 2 that split over a purely inseparable extension E of F of exponent ≥2 but that do not split over any subextension of F inside E of lower exponent. For n≥3, we also give examples of such Pfister forms that split over a purely inseparable extension E of F of degree ≥4 but that do not split over any simple subextension of F inside E. As a consequence, we get a negative answer to a question posed by Bernhard Mühlherr and Richard Weiss who asked whether an octonion division algebra over a field of characteristic 2 that splits over a purely inseparable extension will already split over some quadratic subextension. We use the fact that the splitting of such an algebra is controlled by the splitting of the associated norm form which in turn is given by a 3-fold Pfister form.

Alternative rings whose associators are not zero-divisors

The purpose of this short note is to prove that if R is an alternative ring whose associators are not zero-divisors, then R has no zero-divisors. By a result of Bruck and Kleinfeld, if, in addition, the characteristic of R is not 2, then the central quotient of R is an octonion division algebra over some field.

A short characterization of the octonions

In this article, we prove that if R is a proper alternative ring whose additive group has no 3-torsion and whose non-zero commutators are not zero-divisors, then R has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of R is not 2, then the central quotient of R is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where has 3-torsion.

Factoring octonion polynomials

We provide an analogue of Wedderburn’s factorization method for central polynomials with coefficients in an octonion division algebra, and present an algorithm for fully factoring polynomials of degree [Formula: see text] with [Formula: see text] conjugacy classes of roots, counting multiplicities.

The Unified Standard Model

The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8,$\mathbb{C})$ of complex 8$\times$8 matrices. This algebra is well motivated by its close relation to the normed division algebra of octonions. (Anti-)particle states are identified with basis elements of the vector space M(8,$\mathbb{C})$. Gauge transformations are simply described by the algebra acting on itself. Our result shows that all particles and gauge structures of the Standard Model are contained in the tensor product of all four normed division algebras, with the quaternions providing the Lorentz representations. Interestingly, the space M(8,$\mathbb{C})$ contains two additional elements independent of the Standard Model particles, hinting at a minimal amount of new physics.

Yang–Mills-like field theories built on division quaternion and octonion algebras

Yang–Mills-like field theories are developed using both division quaternion and octonion algebras. Quaternionic field theory is the same as the Yang–Mills gauge theory. The octonionic field theory is an expansion from a similar field theory built with split octonions algebra that uses a complex-form basis as proposed by Gunaydin and Gursey (J Math Phys 14:1651, 1973), now with division octonions. This proposed octonionic field theory is the last possible field theory built using the division algebra of octonions, as permitted by the Hurwitz theorem. The interpretation of this “octonionic field” is not straightforward, except that it is similar to the quaternionic Yang–Mills field. A possible interpretation for this octonionic field theory for spin-1/2 particles is proposed in terms of quantum mechanics.

Polynomial equations over octonion algebras

In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].

Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays

This paper discusses octonion-valued neural networks (OVNNs) with leakage delay, time-varying delays, and distributed delays, for which the states, weights, and activation functions belong to the normed division algebra of octonions. The octonion algebra is a nonassociative and noncommutative generalization of the complex and quaternion algebras, but does not belong to the category of Clifford algebras, which are associative. In order to avoid the nonassociativity of the octonion algebra and also the noncommutativity of the quaternion algebra, the Cayley–Dickson construction is used to decompose the OVNNs into 4 complex-valued systems. By using appropriate Lyapunov–Krasovskii functionals, with double and triple integral terms, the free weighting matrix method, and simple and double integral Jensen inequalities, delay-dependent criteria are established for the exponential stability of the considered OVNNs. The criteria are given in terms of complex-valued linear matrix inequalities, for two types of Lipschitz conditions which are assumed to be satisfied by the octonion-valued activation functions. Finally, two numerical examples illustrate the feasibility, effectiveness, and correctness of the theoretical results.

Orthogonal-gradings on $$H^*$$H∗-algebras

We study set-gradings on proper $$H^*$$ -algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary $$H^*$$ -algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary $$H^*$$ -algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative $$H^*$$ -algebra, or a $${\mathbb Z}_2^3$$ -grading of the complex octonions $${\mathbb O}$$ or a non-group grading which is a refinement of the Peirce decomposition of $${\mathbb O}$$ respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.

Clifford parallelisms defined by octonions

We define (left and right) Clifford parallelisms on a seven-dimensional projective space algebraically, using an octonion division algebra. Thus, we generalize the two well-known Clifford parallelisms on a three-dimensional projective space, obtained from a quaternion division algebra. We determine (for both the octonion and quaternion case) the automorphism groups of these parallelisms. A geometric description of the parallel classes is given with the help of a hyperbolic quadric in a Baer superspace, obtained from the split octonion algebra over a quadratic extension of the ground field, again generalizing results that are known for the quaternion case. In contrast to the quaternion case, the orbits of the two Clifford parallelisms under the group of direct similitudes of the norm form of the algebra are non-trivial in the octonion case. The two spaces of parallelisms can be seen as the point sets of two point-line geometries, both isomorphic to the seven-dimensional projective space. Together with the original space, we thus have three versions of this projective space. We introduce a triality between them which is closely related to the triality of the polar space of split type $$\mathrm {D}_4$$ .

An Efficient Method for Solving Equations in Generalized Quaternion and Octonion Algebras

Quaternions often appear in wide areas of applied science and engineering such as wireless communications systems, mechanics, etc. It is known that are two types of non-isomorphic generalized quaternion algebras, namely: the algebra of quaternions and the algebra of coquaternions. In this paper, we present the formulae to pass from a basis in the generalized quaternion algebras to a basis in the division quaternions algebra or to a basis in the coquaternions algebra and vice versa. The same result was obtained for the generalized octonion algebra. Moreover, we emphasize the applications of these results to the algebraic equations and De Moivre’s formula in generalized quaternion algebras and in generalized octonion division algebras.

Moufang sets arising from polarities of Moufang planes over octonion division algebras

For every octonion division algebra $${\mathcal{O}}$$ , there exists a projective plane which is parametrized by $${\mathcal{O}}$$ ; these planes are related to rank two forms of linear algebraic groups of absolute type E 6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F 4, (2) Moufang sets of type 2 E 6, (3) hermitian Moufang sets of type C 4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs over fields of characteristic different from two, whereas case (4) only occurs over fields of characteristic equal to two. The Moufang sets of type 2 E 6 that we obtain in case (2) are exactly those corresponding to linear algebraic groups of type $${^2E^{29}_{6,1}}$$ ; the explicit description of those Moufang sets was not yet known.

On Hexagonal Structures in Higher Dimensional Theories

We analyze the geometrical background under which many Lie groups relevant to particle physics are endowed with a (possibly multiple) hexagonal structure. There are several groups appearing, either as special holonomy groups on the compactification process from higher dimensions, or as dynamical string gauge groups; this includes groups like SU(2), SU(3), G 2, Spin(7), O(8) as well as E 8 and O(32). We emphasize also the relation of these hexagonal structures with the octonion division algebra, as we expect as well eventually some role for octonions in the interpretation of symmetries in High Energy Physics.

ON OCTONIONIC GRAVITY, EXCEPTIONAL JORDAN STRINGS AND NONASSOCIATIVE TERNARY GAUGE FIELD THEORIES

Novel nonassociative octonionic ternary gauge field theories are proposed based on a ternary bracket. This paves the way to the many physical applications of exceptional Jordan Strings/Membranes and Octonionic Gravity. The old octonionic gravity constructions based on the split octonion algebra Os (which strictly speaking is not a division algebra) is extended to the full fledged octonion division algebra O. A real-valued analog of the Einstein–Hilbert Lagrangian [Formula: see text] involving sums of all the possible contractions of the Ricci tensors plus their octonionic-complex conjugates is presented. A discussion follows of how to extract the Standard Model group (the gauge fields) from the internal part of the octonionic gravitational connection. The role of exceptional Jordan algebras, their automorphism and reduced structure groups which play the roles of the rotation and Lorentz groups is also re-examined. Finally, we construct (to our knowledge) generalized novel octonionic string and p-brane actions and raise the possibility that our generalized 3-brane action (based on a quartic product) in octonionic flat backgrounds of 7,8 octonionic dimensions may display an underlying E7, E8 symmetry, respectively. We conclude with some final remarks pertaining to the developments related to Jordan exceptional algebras, octonions, black-holes in string theory and quantum information theory.

Cayley-Dickson algebras and loops

The Cayley-Dickson process is used in connection with square array representations of the Cayley-Dickson algebras. This involves an array operation on square arrays distinct from matrix multiplication. The arrays give a convenient representation of the octonion division algebra and a description of octonion multiplication. The connection between this description of pure octonion multiplication and seven dimensional real space using products related to the commutator and associator of the octonions is extended to the other Cayley-Dickson algebras and the appropriate real vector space.

Seven-sphere and the exceptional N = 7 and N = 8 superconformal algebras

We study realisations of the exceptional non-linear (quadratically generated, or W-type) N = 8 and N = 7 superconformal algebras with Spin(7) and G 2 affine symmetry currents, respectively. Both the N = 8 and N = 7 algebras admit unitary highest-weight representations in terms of a single boson and free fermions in 8 of Spin(7) and 7 of G 2, with the central charges c 8 = 26 5 and c 7 = 5, respectively. Furthermore, we show that the general coset ansätze for the N = 8 and N = 7 algebras naturally lead to the coset spaces SO(8) × U(1)/ SO(7) and SO(7) × U(1)/ G 2, respectively, as the additional consistent solutions for certain values of the central charge. The coset space SO(8)/ SO(7) is the seven-sphere S 7, whereas the space SO(7)/ G 2 represents the sevensphere with torsion, S T 7. The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realisations. We also comment on some possible applications of our results to string theory.

Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton

We construct an octonionic instanton solution to the seven-dimensional Yang-Mills theory based on the exceptional gauge group G 2 which is the automorphism group of the division algebra of octonions. This octonionic has an extension to a solitonic solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to N = 1 space-time supersymmetry in (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.

Orbits of the automorphism group of the exceptional Jordan algebra.

Necessary and sufficient conditions for two elements of a reduced exceptional simple Jordan algebra a to be conjugate under the automorphism group Aut a of a are obtained. It was known previously that if :a is split, then such elements are exactly those with the same minimum polynomial and same generic minimum polynomial. Also, it was known that two primitive idempotents are conjugate under Aut a if and only if they have the same norm class. In the present paper the notion of norm class is extended and combined with the above conditions on the minimum and generic minimum polynomials to obtain the desired conditions for arbitrary elements of :a. In this paper we characterize the orbits of the automorphism group Aut a of a reduced exceptional simple Jordan algebra I = '(03, y) over a field 'D of characteristic not two or three. Since Jacobson [2, Theorem 9] has shown that in case the octonion (Cayley-Dickson) algebra D is split then such an orbit consists of exactly those elements with the same minimum polynomial and the same generic minimum polynomial, we shall restrict our attention to octonion division algebras. In particular, we shall assume 'I' is infinite. Recall that D(03, y) is the Jordan algebra of 3 x 3 symmetric matrices with entries in D with respect to the involution x I> y 1cty, y = diag {Yl, Y2, Y3}, 0 = Yi e (D, and with multiplication x y= I(xy +yx), xy the usual matrix multiplication. An element x E D(03, y) iS of the form (1) x = aieii aik] with aoi E , ai E , i = 1, 2, 3, where (i, j, k) is a cyclic permutation of (1, 2, 3) and a[ij] = yjaeij + yiceji. We have the mapping x i-x# defined by (2) x# = 2 (aak yykn(a?))eii + 2 (y1(ajak) -aia1)[jk] where x is as in (1) and n is the norm on D. If x#& 0 and x# = 0, we say x is of rank one. Also, we write xxy=(x+y)#-x#-y#. For x, ye 3 write T(x,y)=T(x.y) where T(z) is the usual trace. Received by the editors January 27, 1970. AMS 1969 Subject Classifications. Primary 1740.