We give an algebraic construction of standard modules—infinite-dimensional modules categorifying the Poincaré–Birkhoff–Witt basis of the underlying quantized enveloping algebra—for Khovanov–Lauda–Rouquier algebras in all finite types. This allows us to prove in an elementary way that these algebras satisfy the homological properties of an “affine quasihereditary algebra.” In simply laced types these properties were established originally by Kato via a geometric approach. We also construct some Koszul-like projective resolutions of standard modules corresponding to multiplicity-free positive roots.
"Homological properties of finite-type Khovanov–Lauda–Rouquier algebras." Duke Math. J. 163 (7) 1353 - 1404, 15 May 2014. https://doi.org/10.1215/00127094-2681278