: Seminars

Automorphic Lie algebras are a class of infinite-dimensional Lie algebras over the complex field $\mathbb{C}$ that emerged in the context of mathematical physics, and more precisely in the context of integrable systems. They can be thought of as Lie algebras of meromorphic maps (usually with prescribed poles) from a compact Riemann surface $X$ into a finite-dimensional Lie algebra $\mathfrak{g}$, which are equivariant with respect to a finite group $G$ acting on $X$ and on $\mathfrak{g}$, both by automorphisms. We will discuss a classification for $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ and where $X$ is a complex torus. For each case in the classification, 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. This work has been done in collaboration with Vincent Knibbeler and Sara Lombardo.