The Lie Group Structure of Elliptic/Hyperelliptic $\wp$ Functions
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916278694838272 |
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| author | Hayashi, Masahito Shigemoto, Kazuyasu Tsukioka, Takuya |
| author_facet | Hayashi, Masahito Shigemoto, Kazuyasu Tsukioka, Takuya |
| contents | We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$ Sp(2,$\R$)/$\Z_2$ Lie group structure. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic $\wp$ functions have the SO(3,2) $\cong$ Sp(4,$\R$)/$\Z_2$ Lie group structure. Making use of quadratic invariant forms reveals that hyperelliptic $\wp$ functions with genus three have the SO(9,6) Lie group and/or it's subgroup structure. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_06183 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Lie Group Structure of Elliptic/Hyperelliptic $\wp$ Functions Hayashi, Masahito Shigemoto, Kazuyasu Tsukioka, Takuya Exactly Solvable and Integrable Systems Mathematical Physics We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$ Sp(2,$\R$)/$\Z_2$ Lie group structure. For the genus two case, by constructing a quadratic invariant form, we find that hyperelliptic $\wp$ functions have the SO(3,2) $\cong$ Sp(4,$\R$)/$\Z_2$ Lie group structure. Making use of quadratic invariant forms reveals that hyperelliptic $\wp$ functions with genus three have the SO(9,6) Lie group and/or it's subgroup structure. |
| title | The Lie Group Structure of Elliptic/Hyperelliptic $\wp$ Functions |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics |
| url | https://arxiv.org/abs/2211.06183 |