1.Towards Quaternion Quadratic Phase Fourier Transform (arXiv)
Abstract : The quadratic phase Fourier transform QPFT is a neoteric addition to the class of Fourier transforms and embodies a variety of signal processing tools including the Fourier, fractional Fourier, linear canonical, and special affine Fourier transform. In this paper, we generalize the quadratic phase Fourier transform to quaternion valued signals, known as the quaternion QPFT QQPFT. We initiate our investigation by studying the QPFT of 2D quaternionic signals, then we introduce the QQPFT of 2D quaternionic signals. Using the fundamental relationship between the QQPFT and quaternion Fourier transform QFT, we derive the inverse transform and Parseval and Plancherel formulas associated with the QQPFT. Some other properties including linearity, shift and modulation of the QQPFT are also studied. Finally, we formulate several classes of uncertainty principles UPs for the QQPFT, which including Heisenberg type UP, logarithmic UP, Hardys UP, Beurlings UP and Donohon Starks UP. It can be regarded as the first step in the applications of the QQPFT in the real world.
2. Quaternionic Satake equivalence(arXiv)
Abstract : We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the symmetric variety GL_2n/Sp_2n. We explain how these equivalences fit into the general framework of a geometric Langlands correspondence for real groups and the relative Langlands duality conjecture. As an application, we compute the stalks of the IC-complexes for spherical orbit closures in the quaternionic affine Grassmannian and the loop space of GL_2n/Sp_2n. We show the stalks are given by the Kostka-Foulkes polynomials for GL_n but with all degrees doubled
3Application of Generalized Quaternion in Physics. (arXiv)
Abstract : The applications of quaternion in physics are discussed with an emphasis on the elementary particle symmetry and interaction. Three colours of the quark and the quantum chromodynamics (QCD) can be introduced directly from the invariance of basic equations under the quaternion phase transformation (quaternion gauge invariance). The generalized quaternions obey the SU(3) symmetry. QCD is essentially the quantum quaternion dynamics. The further generalization of SU(3) quaternion to G2 octonion is worked out. We demonstrate that the G2 octonion contains seven tri-generator sets of SU(2) symmetry and three of them form SU(3) subgroup. A model of the elementary particle classification and interaction based on octonion gauge theory is proposed. The model unifies the colour and flavour of all particles. It provides a framework for the unified description of four kinds of elementary particles (quarks, leptons, gauge fields and Higgs bosons) and their interactions.