We find the generating function for group tensors contained in the enveloping algebra of each simple compact group of rank three or less. The generating function depends on dummy variables which carry, as exponents, the degrees and representation labels of the tensors; it suggests an integrity basis, a finite number of elementary tensors, in terms of which all can be expressed as stretched tensor products. We show how the generating functions for tensors in the enveloping algebra of SO(5) and SU(3) reduce when the tensors are acting on the basis of representations for which one of the Cartan labels vanish. The missing label problem in the reduction SO(5) (R-HOOK) SO(3) restricted to SO(5) representations of the type (0,(nu)) is considered; the eigenvalues and eigenvectors of a missing label operator are found up to (including) representation (0,12).

Discrete shapes can be described and analyzed using Lie groups, which are mathematical structures having both algebraic and geometrical properties. These structures, borrowed from mathematical physics, are both algebraic groups and smooth manifolds. A key property of a Lie group is that a curved space can be studied, using linear algebra, by local linearization with an exponential map. Here, a discrete shape was a Euclidean-invariant computer representation of an object. Highly variable shapes are known to exist in non-linear spaces where linear analysis tools, such as Pearson's decomposition of principal components, are inadequate. The novel method proposed herein represented a shape as an ensemble of homogenous matrix transforms. The Lie group of homogenous transforms has elements that both represented a local shape and acted as matrix operators on other local shapes. For the manifold, a matrix transform was found to be equivalent to a vector transform in a linear space. This combination of representation and linearization gave a simple implementation for solving a computationally expensive problem. Two medical datasets were analyzed: 2D contours of femoral head-neck cross-sections and 3D surfaces of proximal femurs. The Lie-group method outperformed the established principal-component analysis by capturing higher variability with fewer components. Lie groups are promising tools for medical imaging and data analysis. / Thesis (Ph.D, Computing) -- Queen's University, 2014-01-30 09:49:03.293

We construct a Lie relator which is not an identical Lie relator. This is the first known example of a non-identical Lie relator. Next we consider the existence of torsion in outer commutator groups. Let L be a free Lie ring. Suppose that 1 < i ≤ j ≤ 2i and i ≤ k ≤ i + j + 1. We prove that L/[L^j, Lⁱ, L^k<./em>] is torsion free. Also, we prove that if 1 < i ≤ j ≤ 2i and j ≤ k ≤ l ≤ i + j then L/[L^j, Lⁱ, L^k, L^l] is torsion free. We then prove that the analogous groups, namely F/[γ_j(F),γ_i(F),γ_k(F)] and F/[γ_j(F),γ_i(F),γ_k(F),γ_l(F)] (under the same conditions for i, j, k and i, j, k, l respectively), are residually nilpotent and torsion free. We prove the existence of 2-torsion in the Lie rings L/[L^j, Lⁱ, L^k] when 1 ≤ k < i,j ≤ 5, and thus show that our methods do not work in these cases. Finally, we consider the order of finite groups of exponent 8. For m ≥ 2, we define the function T(m,n) by T(m,1) = m and T(m,k + 1) = m^T(m,k). We prove that if G is a finite m-generator group of exponent 8 then |G| ≤ T(m, 7⁴⁷¹), improving upon the best previously known bound of T(m, 8⁸⁸).

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 74-76). Available also in a digital version from Dissertation Abstracts.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2009. / Includes bibliographical references (p. 47-48).

Lie Groups occur in math and physics as representations of continuous symmetries and are often described in terms of their Lie Algebra. This thesis is concerned with finding a concrete description of a Lie group given its associated Lie algebra. Several calculations toward this end are developed and then implemented in the Maple Differential Geometry package. Examples of the calculations are given.

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published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy