Reflection functors on quiver Hecke algebras
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
TL;DR
This work extends Lusztig's braid symmetries to quiver Hecke algebras of arbitrary symmetrizable Kac–Moody type by constructing reflection functors. The authors enlarge the underlying algebras with adjacent vertices, realize graded localizations ${\mathscr{A}_\pm}$ via non-degenerate braiders, and employ Schur–Weyl duality data from affinizations to define exact functors that categorify braid actions. They build an explicit pair of reciprocal functors $\mathscr{F}_i$ and $\mathscr{F}_i^*$, restricting to subcategories to yield braid-categorification at the Grothendieck level, thereby completing the program for all finite types. The results provide new algebraic tools for categorification of quantum group braid symmetries and broaden the applicability of reflection functors beyond the finite-type setting. Overall, the paper delivers a robust, algebraic construction of braid symmetry-categorifying functors for quiver Hecke algebras of arbitrary symmetrizable type, with potential impact on representation theory and categorification.
Abstract
We construct the reflection functors for quiver Hecke algebras of an arbitrary symmetrizable Kac-Moody type. These reflection functors categorify Lusztig's braid symmetries.