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Explore the final part of a three-part lecture series on abelian Hall categories delivered by Sabin Cautis from the University of British Columbia. Delve into the construction of finite length abelian categories associated with quivers that categorify corresponding K-theoretic Hall algebras. Learn how the simple objects in these categories provide a dual canonical basis for Hall algebras, with particular emphasis 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 and gain insights into the sophisticated mathematical structures that bridge representation theory, algebraic geometry, and quantum algebra through this advanced mathematical exposition.
