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- Title
Automorphic Lie Algebras and Modular Forms.
- Authors
Knibbeler, Vincent; Lombardo, Sara; Veselov, Alexander P
- Abstract
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let |$\Gamma $| be a finite index subgroup of |$\textrm {SL}(2,\mathbb Z)$| with an action on a complex simple Lie algebra |$\mathfrak g$| , which can be extended to |$\textrm {SL}(2,{\mathbb {C}})$|. We show that the Lie algebra of the corresponding |$\mathfrak {g}$| -valued modular forms is isomorphic to the extension of |$\mathfrak {g}$| over the usual modular forms. This establishes a modular analogue of a well-known result by Kac on twisted loop algebras. The case of principal congruence subgroups |$\Gamma (N), \, N\leq 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.
- Subjects
LIE algebras; DIOPHANTINE equations; MODULAR groups; REPRESENTATIONS of algebras; MODULAR forms; C*-algebras; ALGEBRA; AUTOMORPHIC functions
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 6, p5209
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnab376