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Geiss-Leclerc-Schroer [Invent. Math. 209 (2017)] has introduced a notion of generalized preprojective algebra associated with a generalized Cartan matrix and its symmetrizer. This class of algebra realizes a crystal structure on the set of maximal dimensional irreducible components of the nilpotent variety [Selecta Math. (N.S.) 24 (2018)]. For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirkovic-Vilonen polytopes from generic modules of these components, and give a identification as crystals between the set of Mirkovic-Vilonen polytopes and the set of maximal dimensional irreducible components except for type $mathsf{G}_2$. This generalizes results of Baumann-Kamnitzer [Represent. Theory 16 (2012)] and Baumann-Kamnitzer-Tingley [Publ. Math. Inst. Hautes Etudes Sci. 120 (2014)].
We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiss-Leclerc-Schr{o}er~[Invent.~math.
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schroer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Pi$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We
Let $U_q(mathfrak{g})$ be a quantum affine algebra of arbitrary type and let $mathcal{C}_{mathfrak{g}}$ be Hernandez-Leclercs category. We can associate the quantum affine Schur-Weyl duality functor $F_D$ to a duality datum $D$ in $mathcal{C}_{mathfr
We determine the Z-module structure of the preprojective algebra and its zeroth Hochschild homology, for any non-Dynkin quiver (and hence the structure working over any base commutative ring, of any characteristic). This answers (and generalizes) a c
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine ADE type and let $mathcal{C}^0_{mathfrak{g}}$ be Hernandez-Leclercs category. For a duality datum $mathcal{D}$ in $mathcal{C}^0_{mathfrak{g}}$, we denote by $mathcal{F}_{mathcal{D