论文标题
自态谎言代数和模块化形式
Automorphic Lie algebras and modular forms
论文作者
论文摘要
我们介绍并研究了与模块化组相关的自动层谎言代数的某些双曲线版本。让$ \ $γ$成为$ \ mathrm {sl}（2，\ m athbb {z}）$的有限索引子组，并在复杂的简单lie代数$ \ mathfrak g $上进行操作，可以将其扩展到$ \ mathrm {slrm {slrm {Slrm {sl}（2，\ mathbb {c} c}）$。我们表明，相应的$ \ mathfrak {g} $ - 有价值的模块化表格的谎言代数与通常的模块化表单上的$ \ mathfrak {g} $的扩展是同构的。这建立了KAC在扭曲环代数上的众所周知结果的模块化类似物。有关Klein和Fricke的经典结果以及著名的Markov Diophantine方程，考虑了更多的详细信息中考虑了主要一致性子组$γ（n），\，n \ leq 6 $的情况。我们简要讨论了这些代数的扩展和表示。
We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $Γ$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak g$, which can be extended to $\mathrm{SL}(2,\mathbb{C})$. 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 $Γ(N), \, N\leq 6$ are considered in more details 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.
