must be the field, two copies of the field, the split quaternions, or the split octonions. There is a natural q-version of the composition property that the algebra Oq of quantum octonions is shown to satisfy (see Prop. 4.12 below). We also prove that the quantum octonion algebra Oq satisfies the “q-Principle of Local Triality” (Prop. 3.12). Inside the quantum octonions are two nonisomorphic 4-...

Many quaternions with the coefficients selected from special integer sequences such as Fibonacci and Lucas have been investigated by a great number of researchers. This article presents new classes whose components are composed symmetrical hyperbolic functions. In addition, Binet's formulas, certain generating matrices, functions, Cassini's d'Ocagne's identities for these given.

We relate the moduli space of Yang-Mills instantons to quaternionic manifolds. For instanton number one, the Wolf spaces play an important role. We apply these ideas to instanton calculations in N = 4 SYM theory. Instantons in Yang-Mills theories are defined by the solutions to the self-duality equation in four-dimensional Euclidean space, Fμν = Fμν = 1 2ǫμνρσFρσ , (1) and are characterized by ...

• Rotation matrices: This is the most general form for representing attitude of a body, also called special orthogonal group SO(3) group, is the space of 3×3 matrices satisfying some constraints. Since it uses nine numbers to represent three angular degrees of freedom, there are six independent constraints on the matrix elements. Each column(and row) is unit vector, which gives us 3 constraints...

We investigate doubled (generalized) complex structures in $2D$-dimensional Born geometries where T-duality symmetry is manifestly realized. show that K\"{a}hler, hyperk\"{a}hler, bi-hermitian and bi-hypercomplex of spacetime are implemented as structures. find the generalized K\"{a}hler (hyperk\"{a}hler) appear subalgebras bi-quaternions split-tetra-quaternions. parts these classified by Cliff...