Canonical bases and higher representation theory

Abstract

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, 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 categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case.Comment: 55 pages; DVI may not compile correctly, PDF is preferred. v2: added section on dual canonical bases. v3: improved exposition in line with new version of 1309.3796. v4: final version, to appear in Compositio Mathematica. v5: corrected references for proof of Theorem 4.