Related papers: 3-parameter generalized quaternions
The use of complexified quaternions and $i$-complex geometry in formulating the Dirac equation allows us to give interesting geometric interpretations hidden in the conventional matrix-based approach.
This research aims to define Kac-Moody Lie algebra in Quaternion by using the concept of Quaternification of Lie algebra. The results of this research obtained the definition of Universal Kac-Moody Quaternion Lie algebra, Standard Kac-Moody…
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…
By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators…
Quaternions provide a unified algebraic and geometric framework for representing three-dimensional rotations without the singularities that afflict Euler-angle parametrisations. This article develops a pedagogical and conceptual analysis of…
We introduce and study the algebras of generalized quaternion type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these…
Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an…
Trianguline representations are a certain class of p-adic representations of Gal(Qp^alg/Qp) like the crystalline, semistable and de Rham representations of Fontaine. Their definition involves the theory of (phi,Gamma)-modules. In this…
In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to…
A construction is given of the most general representations of the q-oscillator algebra where both generators are tridiagonal. It is shown to be connected to the Askey-Wilson polynomials.
A parametrization of 3x3 unitary matrices is presented. This mathematical approach is inspired on polarization algebra and is formulated through the identification of a set of three orthonormal three-dimensional Jones vectors representing…
Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…
We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe…
We describe arbitrary multiplicative differential forms on Lie groupoids infinitesimally, i.e., in terms of Lie algebroid data. This description is based on the study of linear differential forms on Lie algebroids and encompasses many known…
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…
Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $\xi_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra…
This is an introduction to advanced linear algebra, with emphasis on geometric aspects, and with some applications included too. We first review basic linear algebra, notably with the spectral theorem in its general form, and with the…
Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras…
The functional space of biquaternions is considered on Minkovskiy space. The scalar-vector biquaternions representation is used which was offered by W. Hamilton for quaternions. With introduction of differential operator - a mutual complex…
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a…