2024/11/14 by Yixing Lan, Lan, Yixin, Yumeng Wu +3
Mathematics · Medicine · #Algebraic structures and combinatorial models #Advanced Algebra and Geometry #Neurosurgical Procedures and Complications
paper · pdf · doi:10.48550/arxiv.2411.09188
This paper continues the work of \citefang2023lusztigsheavesintegrablehighest and \citefang2023lusztigsheavestensorproducts. For a symmetrizable generalized Cartan matrix C and the corresponding quantum group U, we consider an associated quiver Q equipped with an admissible automorphism a. We construct a category \widetildeQ/N obtained from localizations of Lusztig sheaves for the corresponding framed and 2-framed quivers with automorphism. The Grothendieck groups of these categories realize the integrable highest weight module L(λ) and the tensor product L(λ1)⊗ L(λ2) of integrable highest weight U-modules. After quotienting by traceless objects, Lusztig sheaves yield the signed canonical bases of L(λ) and L(λ1)⊗ L(λ2). As applications, we recover symmetrizable crystal structures on Nakajima quiver varieties, Nakajima tensor product varieties, and Lusztig nilpotent varieties of preprojective algebras.