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Automorphic Lie algebras and modular forms
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let Γ be a finite index subgroup of SL(2,Z) with an action on a complex simple Lie algebra g, which can be extended to SL(2,C). We show that the Lie algebra of the corresponding g-valued mo...
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| Format: | Default Article |
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2022
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| Online Access: | https://hdl.handle.net/2134/22100855.v1 |
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| _version_ | 1818165469560963072 |
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| author | Vincent Knibbeler Sara Lombardo Alexander Veselov |
| author_facet | Vincent Knibbeler Sara Lombardo Alexander Veselov |
| author_sort | Vincent Knibbeler (4742670) |
| collection | Figshare |
| description | We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let Γ be a finite index subgroup of SL(2,Z) with an action on a complex simple Lie algebra g, which can be extended to SL(2,C). We show that the Lie algebra of the corresponding g-valued modular forms is isomorphic to the extension of g over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups Γ(N),N≤6, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [See full text paper for actual mathematical formulae.] |
| format | Default Article |
| id | rr-article-22100855 |
| institution | Loughborough University |
| publishDate | 2022 |
| record_format | Figshare |
| spelling | rr-article-221008552022-02-09T00:00:00Z Automorphic Lie algebras and modular forms Vincent Knibbeler (4742670) Sara Lombardo (4425862) Alexander Veselov (1259028) Pure mathematics Science & Technology Physical Sciences Mathematics math.RT General Mathematics Integrable Representations <p>We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let Γ be a finite index subgroup of SL(2,Z) with an action on a complex simple Lie algebra g, which can be extended to SL(2,C). We show that the Lie algebra of the corresponding g-valued modular forms is isomorphic to the extension of g over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups Γ(N),N≤6, is considered in more detail in relation to the classical results of Klein and Fricke and the celebrated Markov Diophantine equation. We finish with a brief discussion of the extensions and representations of these Lie algebras. [See full text paper for actual mathematical formulae.]</p> 2022-02-09T00:00:00Z Text Journal contribution 2134/22100855.v1 https://figshare.com/articles/journal_contribution/Automorphic_Lie_algebras_and_modular_forms/22100855 CC BY 4.0 |
| spellingShingle | Pure mathematics Science & Technology Physical Sciences Mathematics math.RT General Mathematics Integrable Representations Vincent Knibbeler Sara Lombardo Alexander Veselov Automorphic Lie algebras and modular forms |
| title | Automorphic Lie algebras and modular forms |
| title_full | Automorphic Lie algebras and modular forms |
| title_fullStr | Automorphic Lie algebras and modular forms |
| title_full_unstemmed | Automorphic Lie algebras and modular forms |
| title_short | Automorphic Lie algebras and modular forms |
| title_sort | automorphic lie algebras and modular forms |
| topic | Pure mathematics Science & Technology Physical Sciences Mathematics math.RT General Mathematics Integrable Representations |
| url | https://hdl.handle.net/2134/22100855.v1 |