2025 Theses Doctoral

LMN maps in equivariant 𝐾-theory and applications

Lazowski, Davis Michael

The goal of this thesis is to explain various Weyl/braid group actions on categories and characters associated to finite type quantum affine algebras geometrically.

Specifically, we define an interesting action of a braid group on equivariant 𝐾-theory of a finite type ADE Nakajima variety, which we prove is related to the classical Lusztig and Chari braid group actions.

We then combine this braid group action with Henry Liu’s study of asymptotic modules of quantum affine algebras.

As a result, we get a geometric perspective on various formulas and conjectures about TQ relations obtained in the extremely interesting recent works of Frenkel, Hernandez and Wang. All results of this work involving asymptotic algebras rely on conjectural properties of the critical 𝐾-theory of Nakajima varieties. We explain the assumptions further in the main text.