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


Beliakova, Anna; Habiro, Kazuo; Lauda, Aaron D; Živković, Marko (2017). Trace decategorification of categorified quantum $\mathfrak {sl}_2$. Mathematische Annalen, 367(1-2):397-440.

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 $\mathrm{U}(\mathfrak{sl}_2[\mathit{t}])$.

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 $\mathrm{U}(\mathfrak{sl}_2[\mathit{t}])$.

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Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2017
Deposited On:12 Jan 2017 07:38
Last Modified:19 Feb 2018 07:20
Publisher:Springer
ISSN:0025-5831
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s00208-016-1389-y

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