Canonical bases and higher representation theory
We show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a 2-category categorifying the universal enveloping algebra, when the associated Lie algebra is finite type and simply laced. The 2-category is a variation on those defined by Rouquier and Khovanov-Lauda described in a recent paper of Cautis and Lauda. Furthermore, we introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This is a natural generalization of a similar construction of categories for tensor products of highest weight integrable representations from the authors previous work. More generally, we study the theory of bases arising from indecomposable objects in higher representation theory, which we term "orthodox" and the overlap of this theory with the more classical theory of canonical bases.