Indecomposable objects and Lusztig's canonical basis

Abstract

We compute the indecomposable objects of U+3 - The categorification of U+q(sl3), the positive half of quantum sl3 - And we decompose an arbitrary object into indecomposable ones. On the decategorified level, we obtain Lusztig's canonical basis of U+q (sl3). We also categorify the higher quantum Serre relations in U+q(sl3), by defining a certain complex in the homotopy category of U+3 that is homotopic to zero. All our work is done over the ring of integers. This paper is based on the extended diagrammatic calculus introduced to categorify quantum groups.