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Steinberg algebra and Leavitt path algebras
Leavitt path algebras are a new and exciting subject in noncommutative ring theory. To each directed graph E, and unital commutative ring R, is associated an R-algebra called the Leavitt path algebra of E with coefficients in R. It was discovered, when the theory of Leavitt path algebras was already...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2019
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| _version_ | 1867613206110797824 |
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| access_status_str | Open Access |
| author | Rigby, Simon W |
| author2 | Sanchez-Ortega, Juana |
| author_browse | Rigby, Simon W Sanchez-Ortega, Juana |
| author_facet | Sanchez-Ortega, Juana Rigby, Simon W |
| author_sort | Rigby, Simon W |
| collection | Thesis |
| description | Leavitt path algebras are a new and exciting subject in noncommutative ring theory. To each directed graph E, and unital commutative ring R, is associated an R-algebra called the Leavitt path algebra of E with coefficients in R. It was discovered, when the theory of Leavitt path algebras was already quite advanced, that some of the more difficult questions were susceptible to a new approach using topological groupoids. Taking a special kind of groupoid G, one can construct an R-algebra called the Steinberg algebra of G. Many interesting classes of algebras, including Leavitt path algebras, can be obtained from this process. This dissertation is an exposition of the recent advances achieved by the groupoid approach to Leavitt path algebras. New proofs are presented to show that the boundary path groupoid (which underlies the Steinberg algebra model for Leavitt path algebras) has the necessary topological properties. A new theorem is presented, characterising strongly graded Leavitt path algebras in graphical terms. We show that the main results on the structure theory of Leavitt path algebras, including the simplicity and primitivity theorems, can be recovered using the groupoid approach. We demonstrate how these methods lead to an explicit description of the centre of a Leavitt path algebra. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/29433 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:32:27.580Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| 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:11427/29433 Steinberg algebra and Leavitt path algebras Rigby, Simon W Sanchez-Ortega, Juana Mathematics Leavitt path algebras are a new and exciting subject in noncommutative ring theory. To each directed graph E, and unital commutative ring R, is associated an R-algebra called the Leavitt path algebra of E with coefficients in R. It was discovered, when the theory of Leavitt path algebras was already quite advanced, that some of the more difficult questions were susceptible to a new approach using topological groupoids. Taking a special kind of groupoid G, one can construct an R-algebra called the Steinberg algebra of G. Many interesting classes of algebras, including Leavitt path algebras, can be obtained from this process. This dissertation is an exposition of the recent advances achieved by the groupoid approach to Leavitt path algebras. New proofs are presented to show that the boundary path groupoid (which underlies the Steinberg algebra model for Leavitt path algebras) has the necessary topological properties. A new theorem is presented, characterising strongly graded Leavitt path algebras in graphical terms. We show that the main results on the structure theory of Leavitt path algebras, including the simplicity and primitivity theorems, can be recovered using the groupoid approach. We demonstrate how these methods lead to an explicit description of the centre of a Leavitt path algebra. 2019-02-08T14:00:01Z 2019-02-08T14:00:01Z 2018 2019-02-07T09:25:06Z Master Thesis Masters MSc http://hdl.handle.net/11427/29433 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Rigby, Simon W Steinberg algebra and Leavitt path algebras |
| thesis_degree_str | Master's |
| title | Steinberg algebra and Leavitt path algebras |
| title_full | Steinberg algebra and Leavitt path algebras |
| title_fullStr | Steinberg algebra and Leavitt path algebras |
| title_full_unstemmed | Steinberg algebra and Leavitt path algebras |
| title_short | Steinberg algebra and Leavitt path algebras |
| title_sort | steinberg algebra and leavitt path algebras |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/29433 |
| work_keys_str_mv | AT rigbysimonw steinbergalgebraandleavittpathalgebras |