Categorical action of the extended braid group of affine type A

TL;DR

This paper constructs a faithful categorical action of the extended braid group of affine type A using a cyclic quiver algebra, linking algebraic structures with topological intersection numbers.

Contribution

It introduces a new trigraded algebraic framework to realize the extended affine braid group actions categorically.

Findings

Faithful categorical action constructed on the homotopy category

Identification of morphism dimensions with topological intersection numbers

Use of a trigraded cyclic quiver algebra

Abstract

Using a quiver algebra of a cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type A on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.