Universität zu Köln

Schumann, Beatrix Caroline (2014). Homological description of crystal structures on quiver varieties. PhD thesis, Universität zu Köln.

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Abstract

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra and the irreducible highest weight representations of (almost all) simply-laced Lie algebras, respectively. The first realization we consider is a geometric construction in terms of irreducible components of certain quiver varieties established by Kashiwara and Saito. The second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke using Ringel's Hall algebra approach to quantum groups. We connect the two constructions by studying certain sufficiently generic representations of the preprojective algebra. We further show that, in the type $A$ situation, the crystal isomorphism can be described on the combinatorial level via semistandard Young tableaux.