Related papers: Angular Momentum, Quaternion, Octonion, and Lie-Su…
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…
This article presents the derivation of a comprehensive formula for the Clebsch-Gordan coefficients in a quantum system. The formula is derived by employing the iterative application of angular momentum ladder operators on each defined…
Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division…
Using elementary linear algebra, this paper clarifies and proves some concepts about a recently introduced octonion-like associative division algebra over R. This octonion-like algebra is actually the same as the split-biquaternion algebra,…
In this paper we present new formulas, which represent commutators and anticommutators of Clifford algebra elements as sums of elements of different ranks. Using these formulas we consider subalgebras of Lie algebras of pseudounitary…
We interpret the elements of the exceptional Lie algebra $\mathfrak{e}_{8(-24)}$ as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra…
The octonions are the largest of the four normed division algebras. While 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…
There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral…
We construct an explicit example of a generalized Lie 3-algebra from the octonions. In combination with the result of arXiv:0807.0808, this gives rise to a three-dimensional N=2 Chern-Simons-matter theory with exceptional gauge group G_2…
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…
By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion…
We give a full classification of Lie algebras of specific type in complexified Clifford algebras. These sixteen Lie algebras are direct sums of subspaces of quaternion types. We obtain isomorphisms between these Lie algebras and classical…
Recently proposed supergravity theories in odd dimensions whose fields are connection one-forms for the minimal supersymmetric extensions of anti-de Sitter gravity are discussed. Two essential ingredients are required for this construction:…
We construct quasi-solvable quantum mechanical matrix models by employing two different methods, the one is universal enveloping algebra of Lie superalgebra and the other is N-fold supersymmetry. For the former we examine the q(2) and…
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…
The spinor representation of spin-1/2 states can equally well be mapped to a single unit quaternion, yielding a new perspective despite the equivalent mathematics. This paper first demonstrates a useable map that allows Bloch-sphere…
We define a class of orthosymplectic $osp(m;j|2n;\omega)$ and unitary $sl(m;j|n;\epsilon)$ superalgebras which may be obtained from $osp(m|2n)$ and $sl(m|n)$ by contractions and analytic continuations in a similar way as the special linear,…
The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained…
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…
In this work we represent the $1/2$ Spin particles with complex quaternions using a transformation to 2x2 matrices in order to obtain the Pauli matrices. With this representation we determine the states, rotation operators and the total…