Casper Oelen (Heriot-Watt University) – Elliptic automorphic Lie algebras and integrable systems

Automorphic Lie algebras are infinite-dimensional Lie algebras of meromorphic maps (usually with prescribed poles) from a Riemann surface $X$ into a finite-dimensional Lie algebra $\mathfrak{g}$, equivariant with respect to a group acting by automorphisms on both $X$ and $\mathfrak{g}$. They arise naturally in integrable systems through symmetry reductions of Lax pairs and, from an algebraic viewpoint, provide examples of equivariant map algebras.

In this talk, we discuss some algebraic aspects of elliptic automorphic Lie algebras and their connections to integrable systems. We show that well-known algebras in this context, such as Holod's hidden symmetry algebra of the Landau-Lifshitz equation and the Wahlquist-Estabrook prolongation algebra of the same equation, admit, from the automorphic perspective, a particularly simple description in terms of current algebras. The key tools are normal forms for automorphic Lie algebras, analogous to Chevalley normal forms for finite-dimensional simple Lie algebras. Moreover, these normal forms classify up to Lie algebra isomorphism a class of automorphic Lie algebras associated with complex reductive $\mathfrak{g}$, the Klein four-group acting by translations on a complex torus, and arbitrary finite invariant pole sets. This talk is based on joint work with Sara Lombardo and Vincent Knibbeler.