Canonical bases and higher representation theory

TL;DR

This paper develops a general theory of canonical bases within higher representation theory, demonstrating their natural emergence in categorification and providing new insights into Lusztig's bases and tensor product categories.

Contribution

It introduces a unified framework for canonical bases in categorification and extends previous work to tensor products of representations.

Findings

Lusztig's canonical basis corresponds to classes of indecomposable 1-morphisms in categorification.

Categories are constructed whose Grothendieck groups match tensor products of representations.

The theory applies to finite type, simply laced Lie algebras, generalizing prior results.

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.