Trace decategorification of categorified quantum $\mathfrak {sl}_2$

Abstract

The trace or the 0th Hochschild–Mitchell homology of a linear category $\mathcal{C}$ may be regardedas a kind of decategorification of $\mathcal{C}$. We compute the traces of the two versions $\mathcal{U}$ and $\mathcal{U^*}$ of categorified quantum $\mathrm{U}(\mathfrak{sl}_2)$ introduced by the third author. The trace of $\mathcal{U}$ is isomorphic to the split Grothendieck group $\mathit{K}_0(\mathcal{U})$, and the higher Hochschild–Mitchell homology of $\mathcal{U}$ is zero. The trace of $\mathcal{U^*}$ is isomorphic to the idempotented integral form of the current algebra $\mathrm{U}(\mathfrak{sl}_2[\mathit{t}])$.