Réamonn Ó Buachalla

We present recent progress on noncommutative geometric representations of quantum algebraic objects, such as Drinfeld-Jimbo modules, Nichols algebras, and quantum BGG sequences. The noncommutative geometry underlying these realisations is a q-deformed Dolbault complex for the A-series quantum flag manifolds. This complex is built in a very natural way from Lusztig's quantum root vectors, and is shown to be quite sensitive to the required choice of reduced decompositon of the longest element of the Weyl group. When these constructions are restricted to the quantum Grassmannians, they coincide with prior research on the celebrated Heckenberger-Kolb calculus.