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We classify the automorphic Lie algebras of equivariant maps from a complex torus to $\mathfrak{sl}_2(\mathbb{C})$. For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $\mathrm{PSL}_2({\mathbb{Z}})$ , apart from four cases, which are all isomorphic to Onsager's algebra. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line. [ABSTRACT FROM AUTHOR]

The anti-continuum limit of a monotone variational recurrence relation consists of a lattice of uncoupled particles in a periodic background. This limit supports many trivial equilibrium states that persist as solutions of the model with small coupling. We investigate when a persisting solution generates a so-called lamination and prove that near the anti-continuum limit the collection of laminations of solutions is homeomorphic to the (N�−�1)-dimensional simplex, with N the number of distinct local minima of the background potential. This generalizes a result by Baesens and MacKay on twist maps near an anti-integrable limit. [ABSTRACT FROM AUTHOR]