Faithful ADE braid group actions on triangulated categories

I will speak about my joint work with Yury Volkov where we prove that the action of a generalised braid group on an enhanced triangulated category via spherical twists along an ADE-configuration of n-spherical objects is faithful for any integer n not equal to 1. Although various results on the faithfulness of this action have been obtain over the years (e.g. for type A or n=2, and in some other cases), the approach we present allows us to establish the faithfulness for any ADE diagram, any enhanced triangulated category, and any n not equal to 1 almost simultaneously.