Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917034162388992 |
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| author | Neergård, Kai |
| author_facet | Neergård, Kai |
| contents | A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over $\mathbb{C}$ is presented. It is based on a condition that is both necessary and sufficient for a suitably defined Cartan matrix to be expressible by scalar products in a Euclidean vector space. The sufficiency of this condition makes unnecessary subsequent verification of the existence of a Lie algebra or root system corresponding to each Dynkin diagram by explicit construction. The Dynkin diagrams are selected by examination of an easily calculated sequence of minors of a symmetrised Cartan matrix. These minors are mostly integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_19703 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences Neergård, Kai Mathematical Physics Algebraic Geometry Group Theory Rings and Algebras Representation Theory A novel method of determining which Dynkin diagrams represent simple finite-dimensional Lie algebras over $\mathbb{C}$ is presented. It is based on a condition that is both necessary and sufficient for a suitably defined Cartan matrix to be expressible by scalar products in a Euclidean vector space. The sufficiency of this condition makes unnecessary subsequent verification of the existence of a Lie algebra or root system corresponding to each Dynkin diagram by explicit construction. The Dynkin diagrams are selected by examination of an easily calculated sequence of minors of a symmetrised Cartan matrix. These minors are mostly integers. |
| title | Identifying the simple finite-dimensional Lie algebras over $\mathbb{C}$ by means of simple sequences |
| topic | Mathematical Physics Algebraic Geometry Group Theory Rings and Algebras Representation Theory |
| url | https://arxiv.org/abs/2510.19703 |