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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.04453 |
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| _version_ | 1866911982304624640 |
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| author | Li, Zijia Ye, Ke |
| author_facet | Li, Zijia Ye, Ke |
| contents | This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on $\mathrm{U}_n$, $\mathrm{O}_n(\mathbb{R})$, $\mathrm{O}_{n-1,1}(\mathbb{R})$ and $\mathrm{O}_{n-2,2}(\mathbb{R})$. (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04453 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rational Curves on Real Classical Groups Li, Zijia Ye, Ke Algebraic Geometry Symbolic Computation Group Theory 14H45, 20G20, 26C15, 14L35, 14L30, 70B05 This paper is concerned with rational curves on real classical groups. Our contributions are three-fold: (i) We determine the structure of quadratic rational curves on real classical groups. As a consequence, we completely classify quadratic rational curves on $\mathrm{U}_n$, $\mathrm{O}_n(\mathbb{R})$, $\mathrm{O}_{n-1,1}(\mathbb{R})$ and $\mathrm{O}_{n-2,2}(\mathbb{R})$. (ii) We prove a decomposition theorem for rational curves on real classical groups, which can be regarded as a non-commutative generalization of the fundamental theorem of algebra and partial fraction decomposition. (iii) As an application of (i) and (ii), we generalize Kempe's Universality Theorem to rational curves on homogeneous spaces. |
| title | Rational Curves on Real Classical Groups |
| topic | Algebraic Geometry Symbolic Computation Group Theory 14H45, 20G20, 26C15, 14L35, 14L30, 70B05 |
| url | https://arxiv.org/abs/2408.04453 |