2015/04/16 by Bea Schumann, Schumann, Bea
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1504.04304
openalex publication_date 2015/04/16 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
Using methods of homological algebra, we obtain an explicit crystal\nisomorphism between two realizations of crystal bases of the lower part of the\nquantized enveloping algebra of (almost all) finite dimensional simply-laced\nLie algebras. The first realization we consider is a geometric construction in\nterms of irreducible components of certain quiver varieties established by\nKashiwara and Saito. The second is a realization in terms of isomorphism\nclasses of quiver representations obtained by Reineke using Ringel's Hall\nalgebra approach to quantum groups. We show that these two constructions are\nclosely related by studying sufficiently generic representations of the\npreprojective algebra.\n