PBW parametrizations and generalized preprojective algebras
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Publication:6353619
DOI10.1016/J.AIM.2021.108144arXiv2011.06524MaRDI QIDQ6353619
Publication date: 12 November 2020
Abstract: Geiss-Leclerc-Schr"oer [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 Mirkovi'c-Vilonen polytopes from generic modules of these components, and give an identification as crystals between the set of Mirkovi'c-Vilonen polytopes and the set of maximal dimensional irreducible components. This generalizes results of Baumann-Kamnitzer [Represent. Theory 16 (2012)] and Baumann-Kamnitzer-Tingley [Publ. Math. Inst. Hautes 'Etudes Sci. 120 (2014)].
Quantum groups (quantized enveloping algebras) and related deformations (17B37) Torsion theories; radicals on module categories (associative algebraic aspects) (16S90) Ring-theoretic aspects of quantum groups (16T20)
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