Full Text Available
Note: Clicking the button above will open the full text document at the original institutional repository in a new window.
The Classical Lie algebras are more simple than they may appear
The purpose of this dissertation is to consider the classical Lie Algebras, namely: so(n, C), sl(n, C) and sp(n, C), n ≥ 2. Our aim will be to prove that if a Lie Algebra L is classical, except for so(2, C) and so(4, C), then it is simple. The classification and analysis will include finding their r...
Saved in:
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Thesis |
| Language: | English |
| Published: |
Department of Mathematics and Applied Mathematics
2021
|
| Subjects: | |
| Tags: |
No Tags, Be the first to tag this record!
|
| _version_ | 1867613176857624576 |
|---|---|
| access_status_str | Open Access |
| author | Brache, Chad |
| author2 | Blackman, Claire |
| author_browse | Blackman, Claire Brache, Chad |
| author_facet | Blackman, Claire Brache, Chad |
| author_sort | Brache, Chad |
| collection | Thesis |
| description | The purpose of this dissertation is to consider the classical Lie Algebras, namely: so(n, C), sl(n, C) and sp(n, C), n ≥ 2. Our aim will be to prove that if a Lie Algebra L is classical, except for so(2, C) and so(4, C), then it is simple. The classification and analysis will include finding their root systems and the associated Dynkin diagrams. The phrase it's the journey that teaches you a lot about your destination applies quite well here, as the bulk of our discussion will be assembling the tools necessary for proving simplicity. We will begin with some linear algebra proving the Primary decomposition theorem and the Cayley-Hamilton Theorem. Following this, we dive into the world of Lie algebras where we look at Lie algebras of dimensions 1, 2 and 3, representations of Lie algebras, weight spaces, Cartan's criteria and the root space decomposition of a Lie algebra L and define the Dynkin diagram and Cartan matrix. This will all culminate and serve as our arsenal in proving that these classical Lie algebras are all rather simple. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/33684 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:31:58.458Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/33684 The Classical Lie algebras are more simple than they may appear Brache, Chad Blackman, Claire Mathematics and Applied Mathematics The purpose of this dissertation is to consider the classical Lie Algebras, namely: so(n, C), sl(n, C) and sp(n, C), n ≥ 2. Our aim will be to prove that if a Lie Algebra L is classical, except for so(2, C) and so(4, C), then it is simple. The classification and analysis will include finding their root systems and the associated Dynkin diagrams. The phrase it's the journey that teaches you a lot about your destination applies quite well here, as the bulk of our discussion will be assembling the tools necessary for proving simplicity. We will begin with some linear algebra proving the Primary decomposition theorem and the Cayley-Hamilton Theorem. Following this, we dive into the world of Lie algebras where we look at Lie algebras of dimensions 1, 2 and 3, representations of Lie algebras, weight spaces, Cartan's criteria and the root space decomposition of a Lie algebra L and define the Dynkin diagram and Cartan matrix. This will all culminate and serve as our arsenal in proving that these classical Lie algebras are all rather simple. 2021-08-03T10:35:10Z 2021-08-03T10:35:10Z 2021 2021-08-02T11:19:14Z Master Thesis Masters MSc http://hdl.handle.net/11427/33684 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science |
| spellingShingle | Mathematics and Applied Mathematics Brache, Chad The Classical Lie algebras are more simple than they may appear |
| thesis_degree_str | Master's |
| title | The Classical Lie algebras are more simple than they may appear |
| title_full | The Classical Lie algebras are more simple than they may appear |
| title_fullStr | The Classical Lie algebras are more simple than they may appear |
| title_full_unstemmed | The Classical Lie algebras are more simple than they may appear |
| title_short | The Classical Lie algebras are more simple than they may appear |
| title_sort | classical lie algebras are more simple than they may appear |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/33684 |
| work_keys_str_mv | AT brachechad theclassicalliealgebrasaremoresimplethantheymayappear AT brachechad classicalliealgebrasaremoresimplethantheymayappear |