Trace Decategorification of Categorified Quantum sl(3)

Abstract

We prove that the trace of categorified quantum \(\mathfrak {sl}_3\) introduced by Khovanov and Lauda can also be identified with quantum \(\mathfrak {sl}_3\), thus providing an alternative way of decategorification. This is the second step of trace decategorification of quantum \(\mathfrak {sl}_n\) groups over the integers, the first being the \(\mathfrak {sl}_2\) case. The main technique used is decoupling of categorified quantum group into its positive and negative part. This technique can be used for more general categorified quantum groups to reduce the problem to the trace decategorification of its positive part. In the case of quantum \(\mathfrak {sl}_3\), there is an explicit form of the canonical basis of the positive (and isomorphically negative) part of it based on indecomposables found by Stošić, leading to the full result in this case.