In the middle of 1980’s, motivated by study of singularity theories, K. Saito introduced the notion of « elliptic root systems ». Roughly speaking, they are root systems with two null directions. Furthermore, he had classified elliptic root systems $R$ with a one-dimensional subspace of $G$ of two dimensional null directions, under the assumption that the quotient affine root system $R/G$ is reduced. In our joint work with A. Fialowski and K. Iohara, we take off the assumption above, and give the classication of the pair $(R, G)$ with no assumption. In addition, we give an overview of the theory of elliptic root systems in this talk, and certain applications of this theory are also discussed.

https://indico.math.cnrs.fr/event/11537/