Related papers: The Octonions
Octonion algebras over rings are, in contrast to those over fields, not determined by their norm forms. Octonion algebras whose norm is isometric to the norm q of a given algebra C are twisted forms of C by means of the Aut(C)-torsor O(q)…
As is well-known, the real quaternion division algebra $ {\cal H}$ is 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…
The exceptional Lie group E8 plays a prominent role in both mathematics and theoretical physics. It is the largest symmetry group associated with the most general possible normed division algebra, namely, that of the non-associative real…
In this paper, Grand Unified theories are discussed in terms of quaternions and octonions by using the relation between quaternion basis elements with Pauli matrices and Octonions with Gell Mann \lambda matrices. Connection between the…
The purpose of this study is to extend the concept of a generalized Lie $3-$ algebra, known to the divisional algebra of the octonions $\mathbb{O}$, to split-octonions $\mathbb{SO}$, which is non-divisional. This is achieved through the…
It is well-known that the exceptional Lie algebras $\mathfrak{f}_4$ and $\mathfrak{g}_2$ arise from the octonions as the derivation algebras of the $3\times3$ hermitian and $1\times1$ antihermitian matrices, respectively. Inspired by this,…
We show that the octonions are a twisting of the group algebra of Z_2 x Z_2 x 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…
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear…
Answering a question of H. Petersson, we provide a class of examples of pair of octonion algebras over a ring having isometric norms.
We argue that once octonions are formulated as soft Lie algebras, they may be 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)…
We discuss how to represent the non-associative octonionic structure in terms 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…
By applying the idea of viewing the octonions as an associative algebra in certain tensor categories, or more precisely as a twisted group algebra by a 2-cochain, we show that the octonions form an Azumaya algebra in some suitable braided…
In 1934, Jordan et al. gave a necessary algebraic condition, the Jordan identity, for a sensible theory of quantum mechanics. All but one of the algebras that satisfy this condition can be described by Hermitian matrices over the complexes…
The paper surveys recent progress in the search for an appropriate internal space algebra for the Standard Model (SM) of particle physics. As a starting point serve Clifford algebras involving operators of left multiplication by octonions.…
It is shown that the $M$-algebra related with the $M$ theory comes in two variants. Besides the standard $M$ algebra based on the real structure, an alternative octonionic formulation can be consistently introduced. This second variant has…
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…
This paper is devoted to survey composition algebras and some of their applications. After overviewing the classical algebras of quaternions and octonions, both unital composition algebras (or Hurwitz algebras) and symmetric composition…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
Recently, Zhu classified all the SIC-POVMs whose symmetry groups act doubly transitively. Lattices of integers in the complex numbers, the quaternions and the octonions yield the key parts of these symmetry groups.
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…