Kostant's cubic Dirac operator of Lie Superalgebras

We extend equal rank embedding of reductive Lie algebras to that of basic Lie superalgebras. The Kac character formulas for equal rank embedding are derived in terms of subalgebras and Kostant's cubic Dirac operator for equal rank embedding of Lie superalgebras is constructed from both even and odd generators and their related structure constants.Comment: 16 pages, LaTeX2e. To appear in J.Math.Phy

A combination of motion-compensated cone-beam computed tomography iame reconstruction and electrical impedance tomography

Cone-beam computed tomography (CBCT) is an imaging technique used in conjunction with radiation therapy. CBCT is used to verify the position of tumours just prior to radiation treatment session. The accuracy of the radiation treatment of thoracic and upper abdominal tumours is heavily affected by respiratory movement. Blurring artefacts, due to the movement during a CBCT scanning, cause misregistration between the CBCT image and the planning image. There has been growing interest in the use of motion-compensated CBCT for correcting the breathing-induced artefacts. A wide range of iterative reconstruction methods have been developed for CBCT imaging. The direct motion compensation technique has been applied to algebraic reconstruction technique (ART), simultaneous ART (SART), ordered-subset SART (OS-SART) and conjugate gradient least squares (CGLS). In this thesis a dual modality imaging of electrical impedance tomography (EIT) and CBCT is proposed for the first time. This novel dual modality imaging uses the advantages of high temporal resolution of EIT imaging and high spatial resolution of the CBCT method. The main objective of this study is to combine CBCT with EIT imaging system for motion-compensated CBCT using experimental and computational phantoms. The EIT images were used for extracting motion for a motion-compensated CBCT imaging system. A simple motion extraction technique is used for extracting motion data from the low spatial resolution EIT images. This motion data is suitable for input into the direct motion-compensated CBCT. The performance of iterative algorithms for motion compensation was also studied. The dual modality CBCT-EIT is verified using experimental EIT system and computational CBCT phantom data.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Kernel solutions of the Kostant operator on eight-dimensional quotient spaces

After introducing the generators and irreducible representations of the ${\rm su}(5)$ and ${\rm so}(6)$ Lie algebras in terms of the Schwinger's scillators, the general kernel solutions of the Kostant operators on eight-dimensional quotient spaces ${\rm su}(5)/{\rm su}(4)\times {\rm u}(1)$ and ${\rm so}(6)/{\rm so}(4)\times {\rm so}(2)$ are derived in terms of the diagonal subalgebras ${\rm su}(4)\times {\rm u}(1)$ and ${\rm so}(4)\times {\rm so}(2)$, respectively.Comment: 13 pages. Typos correcte

The Chevalley group G_{2}(2) of order 12096 and the octonionic root system of E_{7}

The octonionic root system of the exceptional Lie algebra E_8 has been constructed from the quaternionic roots of F_4 using the Cayley-Dickson doubling procedure where the roots of E_7 correspond to the imaginary octonions. It is proven that the automorphism group of the octonionic root system of E_7 is the adjoint Chevalley group G_2(2) of order 12096. One of the four maximal subgroups of G_2(2) of order 192 preserves the quaternion subalgebra of the E_7 root system. The other three maximal subgroups of orders 432,192 and 336 are the automorphism groups of the root systems of the maximal Lie algebras E_6xU(1), SU(2)xSO(12), and SU(8) respectively. The 7-dimensional manifolds built with the use of these discrete groups could be of potential interest for the compactification of the M-theory in 11-dimension

Cubic interaction vertices for massive and massless higher spin fields

Using the light-cone formulation of relativistic dynamics, we develop various methods for constructing cubic interaction vertices and apply these methods to the study of higher spin fields propagating in flat space of dimension greater than or equal to four. Generating functions of parity invariant cubic interaction vertices for massive and massless higher spin fields of arbitrary symmetry are obtained. We derive restrictions on the allowed values of spins and the number of derivatives, which provide a classification of cubic interaction vertices for totally symmetric fields. As an example of application of the light-cone formalism, we obtain simple expressions for the minimal Yang-Mills and gravitational interactions of massive totally symmetric arbitrary spin fields. We give the complete list of parity invariant and parity violating cubic interaction vertices that can be constructed for massless fields in five and six-dimensional spaces.Comment: 55 pages, LaTeX-2e, v3: Equations (3.15),(3.16) added to Section 3. Discussion of vertices for massless fields in d=4 and footnotes 16,17 added to Section 5.1. New vertices added to Table I. Misprints in equations (7.4), (C.5), and (D.58) correcte

Quaternionic Representation of Snub 24-Cell and its Dual Polytope Derived From E_8 Root System

Vertices of the 4-dimensional semi-regular polytope, \textit{snub 24-cell} and its symmetry group $W(D_{4}):C_{3}$ of order 576 are represented in terms of quaternions with unit norm. It follows from the icosian representation of \textbf{$E_{8}$} root system. The quaternionic root system of $H_{4}$ splits as the vertices of 24-cell and the \textit{snub 24-cell} under the symmetry group of the \textit{snub 24-cell} which is one of the maximal subgroups of the group \textbf{$W(H_{4})$} as well as $W(F_{4})$. It is noted that the group is isomorphic to the\textbf{}semi-direct product of the Weyl group of $D_{4}$ with the cyclic group of order 3 denoted by $W(D_{4}):C_{3}$, the Coxeter notation for which is $[3,4,3^{+}]$. We analyze the vertex structure of the \textit{snub 24-cell} and decompose the orbits of \textbf{$W(H_{4})$} under the orbits of $W(D_{4}):C_{3}$. The cell structure of the snub 24-cell has been explicitly analyzed with quaternions by using the subgroups of the group $W(D_{4}):C_{3}$. In particular, it has been shown that the dual polytopes 600-cell with 120 vertices and 120-cell with 600 vertices decompose as 120=24+96 and 600=24+96+192+288 respectively under the group $W(D_{4}):C_{3}$. The dual polytope of the \textit{snub 24-cell} is explicitly constructed. Decompositions of the Archimedean $W(H_{4})$ polytopes under the symmetry of the group $W(D_{4}):C_{3}$ are given in the appendix.Comment: 20 pages, 5 figure

Comments on the Energy-Momentum Tensor in Non-Commutative Field Theories

In a non-commutative field theory, the energy-momentum tensor obtained from the Noether method needs not be symmetric; in a massless theory, it needs not be traceless either. In a non-commutative scalar field theory, the method yields a locally conserved yet non-symmetric energy-momentum tensor whose trace does not vanish for massless fields. A non-symmetric tensor also governs the response of the action to a general coordinate transformation. In non-commutative gauge theory, if translations are suitably combined with gauge transformations, the method yields a covariantly constant tensor which is symmetric but only gauge covariant. Using suitable Wilson functionals, this can be improved to yield a locally conserved and gauge invariant, albeit non-symmetric, energy-momentum tensor.Comment: LaTeX, 10 pages, no figures; v2: minor changes made, a summary added, version to appear in PL

N=1/2 Global SUSY: R-Matrix Approach

R-matrix method is used to construct supersymmetric extensions of theta - Euclidean group preserving N = 1/2 supersymmetry and its three- parameter generalization. These quantum symmetry supergroups can be considered as global counterparts of appropriately twisted Euclidean superalgebras. The corresponding generalized global symmetry transformations act on deformed superspaces as the usual ones do on undeformed spaces. However, they depend on non(anti)commuting parameters satisfying (anti)commutation relations defined by relevant R matrix.Comment: 30 pages, a number of typos corrected; two references adde