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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2403.15900 |
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| _version_ | 1866914726295896064 |
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| author | Huebschmann, Johannes |
| author_facet | Huebschmann, Johannes |
| contents | This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module. Crossed modules arise from the identities among the relations of the presentation of a group, from the extension problem for groups and, more generally, in low dimensional topology. Also, the (successful) attempt to extend the idea of a normal extension of commutative fields to the realm of non-commutative algebras leads to crossed modules. Crossed modules appear implicitly in a forgotten paper by A. Turing which in principle settles the extension problem for groups. Crossed modules make perfect sense for Lie algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crossed modules Huebschmann, Johannes Group Theory Algebraic Topology 01A60 12G05 18G45 20J05 57M10 This is an overview of the idea of a crossed module. For a group, the triple that consists of the group, its group of automorphisms, and the canonical homomorphism from the group to its group of automorphisms constitutes a crossed module. Crossed modules arise from the identities among the relations of the presentation of a group, from the extension problem for groups and, more generally, in low dimensional topology. Also, the (successful) attempt to extend the idea of a normal extension of commutative fields to the realm of non-commutative algebras leads to crossed modules. Crossed modules appear implicitly in a forgotten paper by A. Turing which in principle settles the extension problem for groups. Crossed modules make perfect sense for Lie algebras. |
| title | Crossed modules |
| topic | Group Theory Algebraic Topology 01A60 12G05 18G45 20J05 57M10 |
| url | https://arxiv.org/abs/2403.15900 |