Octonions and M-theory

We announce a new approach to the octonions as quasiassociative algebras. We strip out the categorical and quasi-quantum group considerations of our longer paper and present here (without proof) some of the more algebraic conclusions

Introducing products between multivectors of Cl(0,7) and octonions, resulting in an octonion, and leading to the non-associative standard octonionic product in a particular case, we generalize the octonionic X-product, associated with the transformation rules for bosonic and fermionic fields on the tangent bundle over the 7-sphere, and the XY-product. We also present the formalism necessary to construct Clifford algebra-parametrized octonions. Finally we introduce a method to...

We present a Veronese formulation of the octonionic and split-octonionic projective and hyperbolic planes. This formulation of the incidence planes highlights the relationship between the Veronese vectors and the rank-1 elements of the Albert algebras over octonions and split-octonions, yielding to a clear formulation of the relationship with the real forms of the Lie groups arising as collineation groups of these planes. The Veronesean representation also provides a novel an...

It is well known that there is a unique $Spin(9)$-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak $Spin(9)$-structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the present article, a new explicit algebraic formula for the $Spin(9)$-invariant 8-form is given. The approach we use generalizes the standard expression of the K\"{a}...

We consider composition and division algebras over the real numbers: We note two r\^oles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.

We propose a non-associative phase space algebra for M-theory backgrounds with locally non-geometric fluxes based on the non-associative algebra of octonions. Our proposal is based on the observation that the non-associative algebra of the non-geometric R-flux background in string theory can be obtained by a proper contraction of the simple Malcev algebra generated by imaginary octonions. Furthermore, by studying a toy model of a four-dimensional locally non-geometric M-theor...

We present a periodic infinite chain of finite generalisations of the exceptional structures, including the exceptional Lie algebra $\mathbf{e_8}$, the exceptional Jordan algebra (and pair) and the octonions. We will also argue on the nature of space-time and indicate how these algebraic structures may inspire a new way of going beyond the current knowledge of fundamental physics.

The theory of representations of Clifford algebras is extended to employ the 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 exhi...

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3+1)-theory (e.g. number of dimensions, existence of maximal velocities, Heisenberg uncertainty, particle generations, etc.). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group G2. This group generates specific rotations of (...

Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history in...