Abelian Hall Categories - Part 1 of 3

Explore the construction of finite length abelian categories associated to quivers that categorify K-theoretic Hall algebras in this mathematical seminar lecture. Learn how these categories provide simple objects that form a dual canonical basis for Hall algebras, with particular focus on affine quivers where this construction yields a basis for the positive half of quantum toroidal algebras. Discover the natural endowment of these abelian categories with renormalized r-matrices, building upon essential background concepts in representation theory and categorical algebra. Gain insights into the deep connections between quiver representations, Hall algebras, and quantum groups through this systematic approach to categorification.