A combinatorial approach to the index of seaweed subalgebras of Kac--Moody algebras

Abstract

In 2000, Dergachev and Kirillov introduced subalgebras of "seaweed type" in $\mathfrak{gl}_n$ and computed their index using certain graphs. Then seaweed subalgebras $\mathfrak q\subset\mathfrak g$ were defined by Panyushev for any reductive $\mathfrak g$. A few years later Joseph generalised this notion to the setting of (untwisted) affine Kac--Moody algebras $\widehat{\mathfrak g}$. Furthermore, he proved that the index of such a seaweed can be computed by the same formula that had been known for $\mathfrak g$. In this paper, we construct graphs that help to understand the index of a seaweed $\mathfrak q\subset\widehat{\mathfrak g}$, where $\widehat{\mathfrak g}$ is of affine type A or C.