Trace decategorification of categorified quantum $$\mathfrak {sl}_2$$ sl 2

Autor: Kazuo Habiro, Anna Beliakova, Aaron D. Lauda, Marko Živković
Rok vydání: 2016
Předmět:
Zdroj: Mathematische Annalen. 367:397-440
ISSN: 1432-1807
0025-5831
DOI: 10.1007/s00208-016-1389-y
Popis: 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 $\dot{\mathcal{U}}^*$ 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 $\mathbf{U}(\mathfrak{sl}_2[t])$.
Databáze: OpenAIRE
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