Utilisateur:Aschheim/Octonions (bib)

of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define $SO(3)$ and $SO(7)$ via conjugation, $SO(4)$ via symmetric multiplication, and $SO(8)$ via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.

Yang-Mills theory based on the exceptional gauge group $G_2$ which is the automorphism group of the division algebra of octonions. This octonionic instanton has an extension to a solitonic two-brane 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 the (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.

 Phys. Lett. B that includes an extra paragraph about the physical properties
 of the octonionic two-brane. We have also put an addendum regarding some

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 Ans"atze for the N=8 and N=7 algebras naturally lead to the coset spaces SO(8)xU(1)/SO(7) and SO(7)xU(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 seven-sphere with torsion, S^7_T. The division algebra of octonions and the associated triality properties of SO(8) play an essential role in all these realizations. We also comment on some possible applications of our results to string theory.

other real division algebras. The pattern of that flexibility is investigated here.

barred operators. This objects give the opportunity to manipulate appropriately the hypercomplex fields. The standard problems arising in the definitions of transpose, determinant and trace for quaternionic and octonionic matrices are immediately overcome. We also investigate the possibility to formulate a new approach to Hypercomplex Group Theory (HGT). From a mathematical viewpoint, our aim is to highlight the possibility of looking at new hypercomplex groups by the use of barred operators as fundamental step toward a clear and complete discussion of HGT.

octonion, in such a manner that it includes the standard complex definition and does not reduce analytic functions to a trivial class. A brief comparison with other definitions is presented.

is quasiconformal in the sense that it leaves invariant a suitably defined ``light cone in 57 dimensions. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E_7 on a 27 dimensional vector space which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned.

lattices, integral domains, sphere fibrations, and other neat stuff.

the highest dimensional Hopf fibration: $S^{7}\longrightarrow S^{15} \longrightarrow S^{8}$.

change the role of the identity. The octonion XY-product is very similar, but shifts the identity as well. This will be of interest to those applying th octonions to string theory.

surprises, which we have reported elsewhere. In particular, the eigenvalues need not be real, there are 6 rather than 3 real eigenvalues, and the corresponding eigenvectors are not orthogonal in the usual sense. The nonassociativity of the octonions makes computations tricky, and all of these results were first obtained via brute force (but exact) Mathematica computations. Some of them, such as the computation of real eigenvalues, have subsequently been implemented more elegantly; others have not. We describe here the use of Mathematica in analyzing this problem, and in particular its use in proving a generalized orthogonality property for which no other proof is known.

spacetime metric signatures. However, the Clifford algebras associated with each are inequivalent, R(4) in the first case (real 4 by 4 matrices), H(2) in the latter (quaternionic 2 by 2). Multivector reformulations of Dirac theory by various authors look quite inequivalent pending the algebra assumed. It is not clear if this is mere artifact, or if there is a right/wrong choice as to which one describes reality. However, recently it has been shown that one can map from one signature to the other using a "tilt transformation" [see P. Lounesto, "Clifford Algebras and Hestenes Spinors", Found. Phys. 23, 1203-1237 (1993)]. The broader question is that if the universe is signature blind, then perhaps a complete theory should be manifestly tilt covariant. A generalized multivector wave equation is proposed which is fully signature invariant in form, because it includes all the components of the algebra in the wavefunction (instead of restricting it to half) as well as all the possibilities for interaction terms.

 on Octonions and Clifford Algebras Algebras, at the 1997 Spring Western
 Sectional Meeting of the American Mathematical Society, Oregon State
 University, Corvallis, OR, 19-20 April 1997.

further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple system in terms of 8-component spinors associated with the Clifford algebras $C(0,7)$ and $C(4,3)$.

matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative non-associative and when they become associative. In particular, these algebras yield special matrix representations of octonions and complex numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix representation of octonions also yields elegant insights into Dirac's equation for a free particle. A few other results and remarks arise as byproducts.

preserving supersymmetry on six- and seven-dimensional manifolds lead to certain generalizations of monopole equations. For six dimensions the string frame metric is Kaehler with the complex structure that descends from the octonions if in addition we assume F^{(1,1)}=0. The susy generator is a gauge covariantly constant spinor. For seven dimensions the string frame metric is conformal to a G_2 metric if in addition we assume the field strength to obey a selfduality constraint. Solutions to these equations lift to geometries of G_2 and Spin(7) holonomy respectively.

is constructed using 3x3 matrices of octonions. The model has both continuum and lattice versions. The lattice version uses HyperDiamond lattice structure.

super-Yang-Mills contains an equal number of bosonic and fermionic non-gauge fields. Besides being manifestly Lorentz and gauge-invariant, this action contains nine spacetime supersymmetries whose algebra closes off-shell. Octonions provide a convenient notation for displaying these symmetries.

octonions, the classical structure of pure mathematics, the 2nd one is Mobilevision, the recently developped technique of computer graphics. Namely, it is shown that the binocular Mobilevision maybe elaborated by use of the octonionic colour space - the 7-dimensional extension of the classical one, which includes a strange overcolour besides two triples of ordinary ones (blue,green, red for left and right eyes).

Contents.
 I. Interpretational geometry, anomalous virtual realities, quantum projective

field theory and Mobilevision:(1.1. Interpretational geometry; 1.2. Anomalous virtual realities; 1.3. Colours in anomalous virtual realities; 1.4. Quantum projective field theory; 1.5. Mobilevision).

 II. Quantum conformal and q_R-conformal field theories, an infinite

dimensional quantum group and quantum field analogs of Euler-Arnold top:(2.1. Quantum conformal field theory; 2.2. Lobachevskii algebra, the quantization of the Lobachevskii plane; 2.3. Quantum q_R-conformal field theory; 2.4. An infinite dimensional quantum group; 2.5. Quantum-field Euler-Arnold top and Virasoro master equation).

 III. Octonionic colour space and binocular Mobilevision:(3.1. Quaternionic

description of ordinary colour space; 3.2. Octonionic colour space and binocular Mobilevision).

geometry of simple (or pure) spinors, as bilinearly underlying euclidean geometry, may be extended also to quantum mechanics of fermions (in first quantization), however in compact momentum spaces, bilinearly constructed with spinors, with signatures unambiguously resulting from the construction, up to sixteen component Majorana-Weyl spinors associated with the real Clifford algebra $\Cl(1,9)$, where, because of the known periodicity theorem, the construction naturally ends. $\Cl(1,9)$ may be formulated in terms of the octonion division algebra, at the origin of SU(3) internal symmetry.

 In this approach the extra dimensions beyond 4 appear as interaction terms in

the equations of motion of the fermion multiplet; more precisely the directions from 5$^{th}$ to 8$^{th}$ correspond to electric, weak and isospin interactions $(SU(2) \otimes U(1))$, while those from 8$^{th}$ to 10$^{th}$ to strong ones SU(3). There seems to be no need of extra dimension in configuration-space. Only four dimensional space-time is needed - for the equations of motion and for the local fields - and also naturally generated by four-momenta as Poincar\'e translations.

 This spinor approach could be compatible with string theories and even

explain their origin, since also strings may be bilinearly obtained from simple (or pure) spinors through sums; that is integrals of null vectors.

satisfies a super-Malcev property and is N=8 supersymmetric. Its Sugawara construction recovers, in a special limit, the non-associative N=8 superalgebra of Englert et al. This paper extends to supersymmetry the results obtained by Osipov in the bosonic case.

are interpreted as equations of motion in momentum spaces, in a constructive approach in which at each step the dimesions of spinor space are doubled while those momentum space increased by two. The construction is possible only in the frame of geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and the momentum spaces result compact, isomorphic toinvariant-mass-spheres imbedded in each other, since the signatures appear to be unambiguously defined and result steadily lorentzian; up to dimension ten with Clifford algebra Cl(1,9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc for multicomponent fermions. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with this spinor geometry, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation. In fact they seem then to be at the origin of electroweak and strong charges, of the 3 families and of the groups of the standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated merely by Poincare translations, and dimensional reduction from Cl(1,9) to Cl(1,3) is equivalent to decoupling of the equations of motion.

division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.

of the associative matrix algebra using the left and right octonionic operators. As an example we construct explicitly some Lie and Super Lie algebra. Then we discuss the notion of octonionic Grassmann numbers and explain its possible application for giving a superspace formulation of the minimal supersymmetric Yang-Mills models.

generalized self-duality in Euclidian eight-dimensions with the original full SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of octonion structure constants made compatible with SO(7) covariance and chirality in 8D. By a simple dimensional reduction together with extra constraints, we derive N=1 supersymmetric self-dual vector multiplet in 7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extra constraints needed on fields and supersymmetry parameter are not obtained from a simple dimensional reduction from 8D. We conjecture that other self-dual supersymmetric theories in lower dimensions D =6 and 5 with respective reduced global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(5) can be obtained in a similar fashion.

constructed using the octonions, or equivalently, triality. An effective continuum physical theory constructed from this model coupled to the balanced model would have a non-vanishing cosmological constant, chiral asymmetry, and a gauge group related to the octonions.

safely used and the non-associativity can be overcame. The necessary points are: (a) Fixing the direction of action by introducing the \delta operator. (b) Closing the \delta algebra by using structure functions f_{ijk} (\phi). (c) Representation of the \delta algebra can be developed. The E or E(\phi) can be found and their structure functions can be computed easily. There may be different applications of soft seven sphere in physics. We have given two cases where the ring division algebras occupies a special position. Self-duality and Simple supersymmetric Yang-Mills theories are two promising places where soft seven sphere prove to be useful and essential.

octonions where its similarities and differences with the quaternionic case are very clear. We use the unified language of Clifford Algebra. We argue that our approach is the pure algebraic formulation of the geometric based soft Lie algebra. The topological criteria for the stability of our solution is given explicitly to establish its solitonic property. Many beautiful features of the parallelizable ring division spheres and Absolute Parallelism (AP) reveal their presence in our formulation.

 new part is added where the incompatibility between the self duality and the

generalization of the usual complex Mobius transformations to the octonions. This is accomplished by relating the octonionic Mobius transformations to the Lorentz group in 10 spacetime dimensions. The result will be of particular interest to physicists working with lightlike objects in 10 dimensions.

Model plus Gravity. The resulting phenomenological model is the D4-D5-E6 model described in hep-ph/9501252 .

D=11 $M$-theory. We consider the octonionic $2^{n+1} \times 2^{n+1}$ Dirac matrices describing the sequence of Clifford algebras with signatures ($9+n,n$) ($n=0,1,2, ...$) and derive the identities following from the octonionic multiplication table. The case $n=1$ ($4\times 4$ octonion-valued matrices) is used for the description of the D=11 octonionic $M$ superalgebra with 52 real bosonic charges; the $n=2$ case ($8 \times 8$ octonion-valued matrices) for the D=11 conformal $M$ algebra with 232 real bosonic charges. The octonionic structure is described explicitly for $n=1$ by the relations between the 528 Abelian O(10,1) tensorial charges $Z_\mu $Z_{\mu\nu}$, $Z_{\mu >... \mu_5}$ of the $M$-superalgebra. For $n=2$ we obtain 2080 real non-Abelian bosonic tensorial charges $Z_{\mu\nu}, Z_{\mu_1 \mu_2 \mu_3}, Z_{\mu_1 ... \mu_6}$ which, suitably constrained describe the generalized D=11 octonionic conformal algebra. Further, we consider the supersymmetric extension of this octonionic conformal algebra which can be described as D=11 octonionic superconformal algebra with a total number of 64 real fermionic and 239 real bosonic generators.

All the exceptional Lie groups, and the projective Cayley line and plane appear in M-theory. Exceptional G_2-holonomy manifolds show up as compactifying spaces, and are related to the M2 Brane and 3-form. We review this evidence, which comes from the initial 11-dim structures. Relations between these objects are stressed, when extant and understood. We argue for the necessity of a better understanding of the role of the octonions themselves (in particular non-associativity) in M-theory.

in an eight dimensional space under an SO(8) gauge field. The underlying mathematics of this particle liquid is that of the last normed division algebra, the octonions. Two fundamentally different liquids with distinct configurations spaces can be constructed, depending on whether the particles carry spinor or vector SO(8) quantum numbers. One of the liquids lives on a 20 dimensional manifold of with an internal component of SO(7) holonomy, whereas the second liquid lives on a 14 dimensional manifold with an internal component of $G_2$ holonomy.

model is constructed that allows computation of force strength constants and constituent mass ratios of elementary particles, with a Lagrangian structure that gives a Higgs scalar particle mass of about 146 GeV and a Higgs scalar field vacuum expectation value of about 252 GeV, giving a tree level constituent Truth Quark (top quark) mass of roughly 130 GeV, which is (in my opinion) supported by dileptonic events and some semileptonic events. See http://galaxy.cau.edu/tsmith/HDFCmodel.html and http://www.innerx.net/personal/tsmith/HDFCmodel.html

 Chapter 1 - Introduction.
 Chapter 2 - From Sets to Clifford Algebras.
 Chapter 3 - Octonions and E8 lattices.
 Chapter 4 - E8 spacetime and particles.
 Chapter 5 - HyperDiamond Lattices.
 Chapter 6 - Internal Symmetry Space.
 Chapter 7 - Feynman Checkerboards.
 Chapter 8 - Charge = Amplitude to Emit Gauge Boson.
 Chapter 9 - Mass = Amplitude to Change Direction.
 Chapter 10 - Protons, Pions, and Physical Gravitons.

representation, in which each element is represented by a left and a right multiplier. This representation can then be used to generate gauge transformations for the purpose of constructing a field theory symmetric under a gauged octonion algebra, the nonassociativity of which appears as a failure of the representation to close, and hence produces new interactions in the gauge field kinetic term of the symmetric Lagrangian.

3-vectors. It is built from two octonionic representations of $E_{8}$, and is reached via $\Lambda_{16}$. It is invariant under the octonion index cycling and doubling maps.

(but equivalent) definitions of matroids, we mention some of the most important theorems of such theory. In particular, we note that every matroid has a dual matroid and that a matroid is regular if and only if it is binary and includes no Fano matroid or its dual. We show a connection between this last theorem and octonions which at the same time, as it is known, are related to the Englert's solution of D = 11 supergravity. Specifically, we find a relation between the dual of Fano matroid and D = 11 supergravity. Possible applications to M-theory are speculated upon.

realizations in terms of either real, complex or quaternionic matrices. Each family includes simple as well as some limiting quasi-simple real Lie algebras. Their relationships naturally lead to an infinite family of $3\times 3$ Freudenthal-like magic squares, which relate algebras in the three CK families. In the lowest dimensional cases suitable extensions involving octonions are possible, and for $N=1, 2$, the "classical" $3\times 3$ Freudenthal-like squares admit a $4\times 4$ extension, which gives the original Freudenthal square and the Sudbery square.

D=4,5 and 7 dimensions as two sequences of superalgebraic structures, taking values in the division algebras R, C and H. The generalized conformal superalgebras are described for D=4 by OSp(1;8| R), for D=5 by SU(4,4;1) and for D=7 by U_\alpha U(8;1|H). The relation with other schemes, in particular the framework of conformal spin (super)algebras and Jordan (super)algebras is discussed. By extending the division-algebra-valued superalgebras to octonions we get in D=11 an octonionic generalized Poincare superalgebra, which we call octonionic M-algebra, describing the octonionic M-theory. It contains 32 real supercharges but, due to the octonionic structure, only 52 real bosonic generators remain independent in place of the 528 bosonic charges of standard M-algebra. In octonionic M-theory there is a sort of equivalence between the octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We also define the octonionic generalized conformal M-superalgebra, with 239 bosonic generators.

numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions. In such a picture physical phenomena are described by the ordinary elements of chosen algebra, while zero divisors (the elements of split-algebras corresponding to zero norms) give raise the coordinatization of space- time. It turns to be possible that two fundamental constants (velocity of light and Planck constant) and uncertainty principle have geometrical meaning and appears from the condition of positive definiteness of norms. The property of non-associativity of octonions could correspond to the appearance of fundamental probabilities in four dimensions. Grassmann elements and a non-commutativity of space coordinates, which are widely used in various physical theories, appear naturally in our approach.

in powers of a complex number $z$ together with a positive sequence of real numbers $\rho(m)=m$. In this article, in analogy with the canonical coherent states, we present a class of vector coherent states by replacing the complex variable $z$ by a real Clifford matrix. We also present another class of vector coherent states by simultaneously replacing $z$ by a real Clifford matrix and $\rho(m)$ by a real matrix. As examples, we present vector coherent states on quaternions and octonions with their real matrix representations.

particular we concentrate over a class of algebras, that we will call self-conjugated, having the property of possessing equivalent right and left multiplication algebras. In this case it is always possible to define an integral sharing many of the properties of the usual integral. For instance, if the algebra has a continuous group of automorphisms, the corresponding derivations are such that the usual formula of integration by parts holds. We discuss also how to integrate over subalgebras. Many examples are discussed, starting with Grassmann algebras, where we recover the usual Berezin's rule. The paraGrassmann algebras are also considered, as well as the algebra of matrices. Since Grassmann and paraGrassmann algebras can be represented by matrices we show also that their integrals can be seen in terms of traces over the corresponding matrices. An interesting application is to the case of group algebras where we show that our definition of integral is equivalent to a sum over the unitary irreducuble representations of the group. We show also some example of integration over non self-conjugated algebras (the bosonic and the $q$-bosonic oscillators), and over non-associative algebras (the octonions).

construct the fundamental and adjoint representations of the octonion algebra. We then show how these representations by associative matrices allow a consistent octonionic gauge theory to be realised. We find that non-associativity implies the existence of new terms in the transformation laws of fields and the kinetic term of an octonionic Lagrangian.

$\Lambda_{24}$, is exposed to further light with this unified and very simple construction.

superconformal algebras and of S{\bf O}(1,9), a modification of SO(1,9) which commutes with $S^7$-transformations. We discuss the relevance of S{\bf O}(1,9) for off-shell super-Maxwell theory in D=(1,9).

X,Y-product variants are used. I present here a general solution for how to go from $G_{2}$ to its X,Y-product variant.

condition in complete analogy to complex analysis is found. The extension to octonions is also worked out.

are developed. We show their applications to a bilocal quark-antiquark or a quark-diquark systems. A new scheme based on SU(6/1) is fully exploited and the bilocal approximation is shown to get carried unchanged into it. Color algebra based on octonions allows the introduction of a new larger algebra that puts quarks, diquarks and exotics in the same supermultiplet as hadrons and naturally suppresses quark configurations that are symmetrical in color space and antisymmetrical in remaining flavor, spin and position variables. A preliminary work on the first order relativistic formulation through the spin realization of Wess-Zumino super-Poincare algebra is presented.

algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

similar manner as our previous construction of the octonions, namely as a twisting of group algebras of Z_2^n by a cocycle. Our approach is more general than the usual one based on generators and relations. We obtain in particular the periodicity properties and a new construction of spinors in terms of left and right multiplication in the Clifford algebra.

homomorphic to the exceptional Lie group G2, is fleshed out by exploring further structures the A-D-E approach of Lie algebraic taxonomy keeps hidden. A breakdown of table equivalence among the half a trillion multiplication schemes the Sedenions allow is found; the 168 elements of PSL(2,7), defining the finite projective triangle on which the Octonions' 480 equivalent multiplication tables are frequently deployed, are shown to give the exact count of primitive unit zero-divisors in the Sedenions. (Composite zero-divisors, comprising all points of certain hyperplanes of up to 4 dimensions, are also determined.) The 168 are arranged in point-set quartets along the 42 Assessors (pairs of diagonals in planes spanned by pure imaginaries, each of which zero-divides only one such diagonal of any partner Assessor). These quartets are multiplicatively organized in systems of mutually zero-dividing trios of Assessors, a D4-suggestive 28 in number, obeying the 6-cycle crossover logic of trefoils or triple zigzags. 3 trefoils and 1 zigzag determine an octahedral vertex structure we call a box-kite -- seven of which serve to partition Sedenion space. By sequential execution of proof-driven production rules, a complete interconnected box-kite system, or Seinfeld production (German for field of being; American for 1990's television's Show About Nothing), can be unfolded from an arbitrary Octonion and any (save for two) of the Sedenions. Indications for extending the results to higher dimensions and different dynamic contexts are given in the final pages.

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.

matrices form two separate families, each containing 3 eigenvalues, and each leading to an orthonormal decomposition of the identity matrix, which would normally correspond to an orthonormal basis. We show here that it nevertheless takes both families in order to decompose an arbitrary vector into components, each of which is an eigenvector of the original matrix; each vector therefore has 6 components, rather than 3.

algebra of the derivations of the split octonions a over an arbitrary field. For this purpose, we will use the Zorn's matrices. We will also compute the multiplication table of this Lie algebra.

representations of the quantum group $U_q$(D$_4$) of D$_4$, we construct a quantum analogue of the split octonions and study its properties. We prove that the quantum octonion algebra satisfies the q-Principle of Local Triality and has a nondegenerate bilinear form which satisfies a q-version of the composition property. By its construction, the quantum octonion algebra is a nonassociative algebra with a Yang-Baxter operator action coming from the R-matrix of $U_q$(D$_4$). The product in the quantum octonions is a $U_q$(D$_4$)-module homomorphism. Using that, we prove identities for the quantum octonions, and as a consequence, obtain at $q = 1$ new ``representation theory proofs for very well-known identities satisfied by the octonions. In the process of constructing the quantum octonions we introduce an algebra which is a q-analogue of the 8-dimensional para-Hurwitz algebra.

Z_2 in the quasitensor category of representations of a quasi-Hopf algebra associated to a group 3-cocycle. We consider general quasi-associative algebras of this type and some general constructions for them, including quasi-linear algebra and representation theory, and an automorphism quasi-Hopf algebra. Other examples include the higher 2^n-onion Cayley algebras and examples associated to Hadamard matrices.

strip out the categorical and quasi-quantum group considerations of our longer paper and present here (without proof) some of the more algebraic conclusions

P(V)$, where $G$ is a complex simple Lie group, $P$ is a maximal parabolic subgroup and $V$ is the minimal $G$-module associated to $P$. Our study began with the observation that Freudenthal's magic chart could be derived from Zak's theorem on Severi varieties and standard geometric constructions. Our attempt to understand this observation led us to discover further connections between projective geometry and representation theory.

 Among other things, we calculate the variety of tangent directions to lines

on $X$ through a point and determine unirulings of $X$. We show this variety is a Hermitian symmetric space if and only if $P$ does not correspond to a short root. We describe the spaces corresponding to the exceptional short roots and their unirulings using the octonions. Further calculations, in the case $X$ is a Hermitian symmetric space, give rise to a strict prolongation property and the appearance of secant varieties at the infinitesimal level. Our work complements and advances that of Freudenthal and Tits, who studied homogeneous varieties in an abstract/axiomatic setting.

(quasialgebras) living in a symmetric monoidal category. In this note we provide further examples of quasialgebras, namely ones where the nonassociativity is induced by a Z_n-grading and a nontrivial 3-cocycle

We show how projection over the real field produces the standard Attiyah-Bott-Shapiro classification.

algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1/2 states, then the same method will lead to the Lie-super algebra osp(1,2). Also, the quantum generalization of the method is discussed on the basis of the quantum group $su_q(2)$.

Cayley-Dickson process beyond the Sedenions are explored. Prior work showed a ZD system in the Sedenions, based on 7 octahedral lattices ("Box-Kites"), whose 6 vertices collect and partition the "42 Assessors" (pairs of diagonals in planes spanned by pure imaginaries, one a pure Octonion, hence of subscript < 8, the other a Sedenion of subscript > 8 and not the XOR with 8 of the chosen Octonion). Potential connections to fundamental objects in physics (e.g., the curvature tensor and pair creation) are suggested. Structures found in the 32-ions ("Pathions") are elicited next. Harmonics of Box-Kites, called here "Kite-Chain Middens," are shown to extend indefinitely into higher forms of 2n-ions. All non-Midden-collected ZD diagonals in the Pathions, meanwhile, are seen belonging to a set of 15 "emanation tables," dubbed "sand mandalas." Showcasing the workings of the DMZ's (dyads making zero) among the products of each of their 14 Assessors with each other, they house 168 fillable cells each (the number of elements in the simple group PSL(2,7) governing Octonion multiplication). 7 of these emanation tables, whose "inner XOR" of their axis-pairs' indices exceed 24, indicate modes of collapsing from higher to lower 2n-ion forms, as they can be "folded up" in a 1-to-1 manner onto the 7 Sedenion Box-Kites. These same 7 also display surprising patterns of DMZ sparsity (with but 72 of 168 available cells filled), with the animation-like sequencing obtaining between these 7 "still-shots" indicating an entry-point for cellular-automata-like thinking into the foundations of number theory.

 mislabeling of standard operator names on p. 4; added paragraph of

formulated on the lightcone of a ten-dimensional, lorentzian, momentum space. Most of the equations for fermion multiplets, usually adopted by particle physics, are then naturally obtained and their properties like internal symmetries, charges, families appear to be due to the correlation of the associated Clifford algebras, with the 3 complex division algebras: complex numbers at the origin of U(1) and charges; quaternions at the origin of SU(2); families and octonions at the origin of SU(3). Pure spinors instead could be relevant not only because the underlying momentum space results compact, but also because it may throw light on some aspects of particle physics, like: masses, charges, constaint relations, supersymmetry and epistemology.

can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.

existence of an approximate hadronic supersymmetry. Effective Hamiltonian of the relativistic quark model is derived, leading to hadronic mass formulae in remarkable agreement with experiments. Bilocal approximation to hadronic structure and incorporation of color through octonion algebra (based on quark-antidiquark symmetry) is also shown to predict exotic diquark-antidiquark ($D-\bar{D}$) meson states. A minimal supersymmetric sceme based on $SU(3)^c \times SU(6/1)$ that excludes exotics is constructed. Symmetries of three quark systems and possible relativistic formulation of the quark model through the spin realization of Wess-Zumino algebra is presented.

twistors and their relation to harmonic analyticity is presented. Generalization of $SL(2,\cal{C})$ to the D=4 conformal group SO(5,1) and its covering group $SL(2,\cal{Q})$ that generalizes the euclidean Lorentz group in $R^4$ [namely $SO(3,1)\approx SL(2,\cal{C})$ which allow us to obtain the projective twistor space $CP^3$] is shown. Quasi-conformal fields are introduced in D=4 and Fueter mappings are shown to map self-dual sector onto itself (and similarly for the anti-self-dual part). Differentiation of Fueter series and various forms of differential operators are shown, establishing the equivalence of Fueter analyticity with twistor and harmonic analyticity. A brief discussion of possible octonion analyticity is provided.

holonomy groups.

   They are classified by (1) real, complex, quaternion or octonion number

they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively.

   For vector bundles over such manifolds, we introduce (special)

A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) A/2-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds.

   We also discuss geometric dualities from this viewpoint: Fourier

transformations on A-geometry for flat tori and a conjectural SYZ mirror transformation from (special) A-geometry to (special) A/2-Lagrangian geometry on mirror special A-manifolds.

of octonions has a large class of exact finite quaternionic power series solutions.

dimensions with the reduced holonomy G_2 \subset SO(7), including all higher-order terms. As its foundation, we first establish N=2 supergravity without self-duality in Euclidean seven dimensions. We next show how the generalized self-duality in terms of octonion structure constants can be consistently imposed on the superspace constraints. We found two self-dual N=2 supergravity theories possible in 7D, depending on the representations of the two spinor charges of N=2. The first formulation has both of the two spinor charges in the {\bf 7} of G_2 with 24 + 24 on-shell degrees of freedom. The second formulation has both charges in the {\bf 1} of G_2 with 16 + 16 on-shell degrees of freedom.

variants. Besides the standard $M$ algebra based on the real structure, an alternative octonionic formulation can be consistently introduced. This second variant has striking features. It involves only 52 real bosonic generators instead of 528 of the standard $M$ algebra and moreover presents a novel and surprising feature, its octonionic $M5$ (super-5-brane) sector is no longer independent, but coincides with the octonionic $M1$ and $M2$ sectors. This is in consequence of the non-associativity of the octonions. An octonionic version of the superconformal $M$-algebra also exists. It is given by $OSp(1,8|{\bf O})$ and admits 239 bosonic and 64 fermionic generators. It is speculated that the octonionic $M$-algebra can be related to the exceptional Lie and Jordan algebras that apparently play a special role in the Theory Of Everything.

as a vector on a Hilbert space, where the Hilbert space is taken to be the tensor product of several other Hilbert spaces. As examples vector coherent states labeled by multiple of quaternions and octonions were given. The resulting generalized oscillator algebra is discussed.