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