Related papers: Unified Octonionic Representation of the 10-13 Dim…
We give a one dimensional octonionic representation of the different Clifford algebra $Cliff(5,5)\sim Cliff(1,9), Cliff(6,6)\sim Cliff(2,10)$ and lastly $Cliff(7,6)\sim Cliff(3,10)$.
A real representation theory of real Clifford algebra has been studied in further detail, especially in connection with Fierz identities. As its application, we have constructed real octonion algebras as well as related octonionic triple…
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…
Octonionic algebra being nonassociative is difficult to manipulate. We introduce left-right octonionic barred operators which enable us to reproduce the associative GL(8,R) group. Extracting the basis of GL(4,C), we establish an interesting…
The connection of (split-)division algebras with Clifford algebras and supersymmetry is investigated. At first we introduce the class of superalgebras constructed from any given (split-)division algebra. We further specify which real…
Quaternionic and octonionic realizations of Clifford algebras and spinors are classified and explicitly constructed in terms of recursive formulas. The most general free dynamics in arbitrary signature space-times for both quaternionic and…
Extended gamma matrix Clifford--Dirac and SO(1,9) algebras in the terms of $8 \times 8$ matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two…
Contemporary presentation of the version 1 demonstrates briefly the development of our investigations and our future goals. The improved free of difficulties in interpretation and printing errors version is presented. The 256-dimensional…
This paper is meant to be an informative introduction to spinor representations of Clifford algebras. In this paper we will have a look at Clifford algebras and the octonion algebra. We begin the paper looking at the quaternion algebra…
The associative Cayley-Dickson algebras over the field of real numbers are also Clifford algebras. The alternative but nonassociative real Cayley-Dickson algebras, notably the octonions and split octonions, share with Clifford algebras an…
Real Clifford algebras play a fundamental role in the eight real Altland-Zirnbauer symmetry classes and the classification tables of topological phases. Here, we present another elegant realization of real Clifford algebras in the…
We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that complex spinors are unavoidably either real Majorana…
We study briefly some properties of real Clifford algebras and identify them as matrix algebras. We then show that the representation space on which Clifford algebras act are spinors and we study in details matrix representations. The…
We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions,…
The well-known classification of the Clifford algebras $Cl(r,s)$ leads to canonical forms of complex and real representations which are essentially unique by virtue of the Wedderburn theorem. For $s\ge 1$ representations of $Cl(r,s)$ on…
The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from…
We present an eight-dimensional even sub-algebra of the ${2^4=16}$-dimensional associative Clifford algebra ${\mathrm{Cl}_{4,0}}$ and show that its eight-dimensional multivectors ${\bf X}$ and ${\bf Y}$ respect the composition law ${||{\bf…
Real Clifford algebras for arbitrary number of space and time dimensions as well as their representations in terms of spinors are reviewed and discussed. The Clifford algebras are classified in terms of isomorphic matrix algebras of real,…
We prove that the alternative Clifford algebra of a nondegenerate ternary quadratic form is an octonion algebra over the ring of polynomials in one variable over the field of definition.
Following [1] we further apply the octonionic structure to supersymmetric 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,…