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    Superalgebras, the Brauer-Wall Group and the Super Frobenius-Schur indicator

    Adriano, Daryl Zane (2023) Superalgebras, the Brauer-Wall Group and the Super Frobenius-Schur indicator. Masters thesis, National University of Ireland Maynooth.

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    In this thesis, we will study the theory of superalgebras, which are algebras with a C2-grading. One of our main aims is to show that many concepts and theorems in Algebra Theory have their counterparts in Superalgebra Theory. For example, we will state and prove the superalgebra counterparts of Schur’s Lemma, Maschke’s Theorem, and Wedderburn’s theorem. In Algebra Theory, each field F has a group called the Brauer Group of F (denoted as Br(F)), which is a group of equivalence classes of central simple F-algebras. We will be showing that there is a superalgebra equivalent, namely the Brauer-Wall group of F (denoted as BW(F)), which is a group of equivalence classes of super central simple F-superalgebras. Additionally, we will be studying group superalgebras, super representations, and super characters. In the study of ordinary group algebras, the Frobenius-Schur indicator meaningfully associates an irreducible C-character of a finite group G with a division algebra over R. In this thesis, we will introduce the Super Frobenius-Schur indicator, which associates a super irreducible C-super character with a super division algebra over R. We will also give the full decomposition of group superalgebras over R and C. Finally, we will discuss Clifford Algebras, another family of examples of superalgebras.

    Item Type: Thesis (Masters)
    Keywords: Superalgebras; Brauer-Wall Group; Super Frobenius-Schur indicator;
    Academic Unit: Faculty of Science and Engineering > Mathematics and Statistics
    Item ID: 17828
    Depositing User: IR eTheses
    Date Deposited: 14 Nov 2023 12:33
      Use Licence: This item is available under a Creative Commons Attribution Non Commercial Share Alike Licence (CC BY-NC-SA). Details of this licence are available here

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