Realization of graded matrix algebras as Leavitt path algebras
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arXiv
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866908374469181440 |
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| author | Vas, Lia |
| author_facet | Vas, Lia |
| contents | While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field, trivially graded by the ring of integers, which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. If $R$ is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which $R$ can be realized as a Leavitt path algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_05174 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Realization of graded matrix algebras as Leavitt path algebras Vas, Lia Rings and Algebras 16W50, 16S50, 16D70 While every matrix algebra over a field $K$ can be realized as a Leavitt path algebra, this is not the case for every graded matrix algebra over a graded field. We provide a complete description of graded matrix algebras over a field, trivially graded by the ring of integers, which are graded isomorphic to Leavitt path algebras. As a consequence, we show that there are graded corners of Leavitt path algebras which are not graded isomorphic to Leavitt path algebras. This contrasts a recent result stating that every corner of a Leavitt path algebra of a finite graph is isomorphic to a Leavitt path algebra. If $R$ is a finite direct sum of graded matricial algebras over a trivially graded field and over naturally graded fields of Laurent polynomials, we also present conditions under which $R$ can be realized as a Leavitt path algebra. |
| title | Realization of graded matrix algebras as Leavitt path algebras |
| topic | Rings and Algebras 16W50, 16S50, 16D70 |
| url | https://arxiv.org/abs/1910.05174 |