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There are four division algebras over $\mathbb{R}$, namely real numbers, complex numbers, quaternions, and octonions. Lack of commutativity and associativity make it difficult to investigate algebraic and geometric properties of octonions.…
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…
It is shown that the supersymmetric quantum mechanics has an octonionic generalization. The generalization is based on the inclusion of quaternions into octonions. The elements from the coset octonions/quaternions are unobservables bacause…
Pauli matrices are 2x2 tracefree matrices with a real diagonal and complex (complex-conjugate) off-diagonal entries. They generate the Clifford algebra Cl(3). They can be generalised by replacing the off-diagonal complex number by one…
The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in…
We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices…
In this paper we introduce a new algebraic device, which enables us to treat the quaternions as though they were a commutative field. This is of interest both for its own sake, and because it can be applied to develop an "algebraic…
In this work we study the recently introduced octonionic duality for membranes. Restricting the self - duality equations to seven space dimensions, we provide various forms for them which exhibit the symmetries of the octonionic and…
We give explicit bijective correspondences between three families of objects: certain pairs of quaternions, which we regard as spinors; certain flags in (1+4)-dimensional Minkowski space; and horospheres in 4-dimensional hyperbolic space…
In this paper, we define three types of 2-Ruled hypersurfaces in the Minkowski 4-space $\mathbb{E}^4_1$. We obtain Gaussian and mean curvatures of the 2-ruled hypersurfaces of type-1 and type-2, and some characterizations about its…
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…
In 1900, Macfarlane proposed a hyperbolic variation on Hamilton's quaternions that closely resembles Minkowski spacetime. Viewing this in a modern context, we expand upon Macfarlane's idea and develop a model for real hyperbolic 3-space in…
The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension $n$. The maximal possible symmetry is realized by the…
The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective…
While real Hamiltonian mechanics and Hermitian quantum mechanics can both be cast in the framework of complex canonical equations, their complex generalisations have hitherto been remained tangential. In this paper quaternionic and…
The 8 $\times$ 8 matrix representation of SO(8) Symmetry has been defined by using the direct product of Pauli matrices and Gamma matrices. These 8 $\times$ 8 matrices are being used to describe the rotations in SO(8) symmetry. The…
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…
The problem of computing the integral cohomology ring of the symmetric square of a topological space has been of interest since the 1930s, but limited progress has been made on the general case until recently. In this work we offer a…
We describe space-time using split octonions over the reals and use their group of automorphisms, the non-compact form of Cartan's exceptional Lie group G2, as the main geometrical group of the model. Connections of the G2-rotations of…
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The…