Twisted quadratic foldings of root systems

The present paper is devoted to twisted foldings of root systems that generalize the involutive foldings corresponding to automorphisms of Dynkin diagrams. A motivating example is Lusztig’s projection of the root system of type E 8 E_8 onto the subring of icosians of the quaternion algebra, which gives the root system of type H 4 H_4 . By using moment graph techniques for any such folding, a map at the equivariant cohomology level is constructed. It is shown that this map commutes with characteristic classes and Borel maps. Restrictions of this map to the usual cohomology of projective homogeneous varieties, to group cohomology and to their virtual analogues for finite reflection groups are also introduced and studied.

Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications

In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable \emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.

Three-phase Quaternion Power in Three-wire Systems

Instantaneous power theory has a central role in power systems analysis. Among mathematical settings used for the development of this theory, quaternion algebra has been used for describing electrical variables in recent works. In this context, this paper aims to describe three-phase power in a quaternion framework. We analyze quaternion power for balanced and unbalanced delta loads, comparing the expressions obtained to the usual expressions of complex power. The quaternion power expression obtained also makes it natural to introduce a decomposition of the unbalanced load in terms of a balanced component and an unbalanced load with null average power. It is also shown that delta unbalanced loads are equivalent to time-varying balanced loads. The results obtained extend the power systems theory in the quaternion domain and emphasize the advantages of using this framework.

New Set of Invariant Quaternion Krawtchouk Moments for Color Image Representation and Recognition

One of the most often used techniques to represent color images is quaternion algebra. This study introduces the quaternion Krawtchouk moments, QKrMs, as a new set of moments to represent color images. Krawtchouk moments (KrMs) represent one type of discrete moments. QKrMs use traditional Krawtchouk moments of each color channel to describe color images. This new set of moments is defined by using orthogonal polynomials called the Krawtchouk polynomials. The stability against the translation, rotation, and scaling transformations for QKrMs is discussed. The performance of the proposed QKrMs is evaluated against other discrete quaternion moments for image reconstruction capability, toughness against various types of noise, invariance to similarity transformations, color face image recognition, and CPU elapsed times.

Counting and equidistribution in quaternionic Heisenberg groups

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

Dual-complex quaternion representation of gravitoelectromagnetism

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.