Trace decategorification of categorified quantum sl(2)

Abstract

The trace or the $0$th Hochschild--Mitchell homology of a linear category $\mathcal{C}$ may be regarded as a kind of decategorification of $\mathcal{C}$. We compute traces of the two versions $\dot{\mathcal{U}}$ and ${\dot{\mathcal{U}}}^{\ast }$ of categorified quantum ${\mathfrak{sl}}_2$ introduced by the third author. One version of the trace coincides with the split Grothendieck group $K_0(\dot{\mathcal{U}})$, which is known to be isomorphic to the the integral idempotented form $\dot{\mathbf{U}}({\mathfrak{sl}}_2)$ of quantum $\mathfrak{sl}(2)$. The higher Hochschild--Mitchell homology in this case is zero. The trace of the second version is isomorphic to the idempotented integral form of the current algebra $\mathbf{U}({\mathfrak{sl}}_2[t])$.

33 pages, xypic figures

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