Browsing by Author "Dr. Aloysius Helminck, Committee Chair"

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    Algorithms for Computing Restricted Root Systems and Weyl Groups
    (2006-05-04) Cicco, Tracey Martine Westbrook; Dr. Aloysius Helminck, Committee Chair; Dr. Ernest Stitzinger, Committee Member; Dr. Tom Lada, Committee Member; Dr. Amassa Fauntleroy, Committee Member
    While the computational packages LiE, Gap4, Chevie, and Magma are sufficient for work with Lie Groups and their corresponding Lie Algebras, no such packages exist for computing the k-structure of a group or the structure of symmetric spaces. My goal is to examine the k-structure of groups and the structure of symmetric spaces and arrive at various algorithms for computing in these spaces.
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    Lifting Automorphisms from Root Systems to Lie Algebras
    (2010-04-30) Watson, Robert Loyd; Dr. Aloysius Helminck, Committee Chair; Dr. Naihuan Jing, Committee Member; Dr. Amassa Fauntleroy, Committee Member; Dr. Ernie L. Stitzinger, Committee Member
    In 1996 and 2000 A.G. Helminck gave the first algorithms for computing some of the structure of symmetric spaces. In this thesis we extend these results by designing algorithms for other aspects of the structure of local symmetric spaces. We begin with an involution on the root system. We would like to understand how this involution describes an involution on the Lie algebra. To do so, we consider the concept of lifting. We say an involution θ on the root system Φ can be lifted to an involution θ on the algebra if we can find θ so that θ|Φ = θ. Success gives rise to a method to compute local symmetric spaces. Accomplishing this task requires effort on multiple fronts. On a small scale we consider a correction vector. A correction vector lives in the toral subalgebra of the Lie algebra. A result due to Steinberg establishes a unique Lie algebra automorphism that can always be defined. We can modify this map with the correction vector so that it becomes an involution. On a large scale, computing the correction vector is too cumbersome. We will show how to “break apart†larger involutions on the root system by projecting the roots into the local symmetric space, then “extracting†specific sub-systems. We can correct the involution on each sub-system, then “glue†the pieces together to form the involution on the whole algebra. This process not only vastly improves the timing of the lifting process, but also gives rise to an argument that any involution on the root system can be lifted. We then present an entire computer package written for Mathematica) for working with local symmetric spaces. This package includes the algorithms we devise, as well as “helper†algorithms which are necessary for implementation.
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    On the classification of orbits of minimal parabolic k-subgroups acting on symmetric k-varieties of SL(n,k)
    (2008-04-25) Beun, Stacy L.; Dr. Tom Lada, Committee Member; Dr. Ernest Stitzinger, Committee Member; Dr. Aloysius Helminck, Committee Chair; Dr. Amassa Fauntleroy, Committee Member
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    Relationship Between Symmetric and Skew-Symmetric Bilinear Forms on V=kn and Involutions of SL(n,k) and SO(n,k,beta)
    (2003-10-22) Dometrius, Christopher; Dr. Aloysius Helminck, Committee Chair; Dr. Naihuan Jing, Committee Member; Dr. Tom Lada, Committee Member; Dr. Ernest Stitzinger, Committee Member
    In this paper, we show how viewing involutions on matrix groups as having been induced by a given non-degenerate symmetric or skew-symmetric bilinear form on the vector space of corresponding dimension can lead to a classification up to isomorphism of the resulting reductive symmetric space in the group setting. We establish a direct link between bilinear algebraic properties of the vector space V=kˆn for an arbitrary field k of characteristic not 2 and involutions of the matrix group G, where G is a subgroup of GL(n,k). Symmetric spaces are defined in terms of involutions, and the development of this classification theory which classifies the involutions also classifies the symmetric spaces coming from these involutions. We classify all involutions on SL(n,k) and develop important foundations for a full classification of the involutons of SO(n,k,beta) where beta is any non-degenerate symmetric bilinear form. We prove that all involutions of SO(n,k,beta) are inner when n is odd. Additionally, we provide criteria for the matrix which gives the conjugation that is the inner involution of SO(n,k,beta), which covers all involutions when n is odd and all involutions which can be written as conjugations when n is even, a fact which is proven in this thesis.