Homological Properties of Standard KLR Modules
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Khovanov-Lauda-Rouquier algebras, or KLR algebras, are a family of algebras known to categorify the upper half of the quantized enveloping algebra of a given Lie algebra. In finite type, these algebras come with a family of standard modules, which correspond to PBW bases under this categorification. In this thesis, we show that there are no non-zero homomorphisms between distinct standard modules and that all non-zero endomorphisms of standard modules are injective. We then apply this result to obtain applications to the modular representation theory of KLR algebras. Restricting our attention to finite type A, we are then able to compute explicit projective resolutions of all standard modules. Finally, in finite type A when alpha is a positive root, we let D be the direct sum of all distinct standard modules and compute the algebra structure on Ext(D, D). This dissertation includes unpublished co-authored material.