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Trace as an alternative decategorification functor


Beliakova, Anna; Guliyev, Zaur; Habiro, Kazuo; Lauda, Aaron D (2014). Trace as an alternative decategorification functor. Acta Mathematica Vietnamica, 39(4):425-480.

Abstract

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called “decategorification” functor. Algebras are typically categorified to additive categories with additional structure, and decategorification is usually given by the (split) Grothendieck group. In this article, we study an alternative decategorification functor given by the trace or the zeroth Hochschild–Mitchell homology. We show that this form of decategorification endows any two representations of the categorified quantum $\mathfrak {sl}_{n}$ with an action of the current algebra $\mathbf {U}(\mathfrak {sl}_{n}[t])$ on its center.

Abstract

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called “decategorification” functor. Algebras are typically categorified to additive categories with additional structure, and decategorification is usually given by the (split) Grothendieck group. In this article, we study an alternative decategorification functor given by the trace or the zeroth Hochschild–Mitchell homology. We show that this form of decategorification endows any two representations of the categorified quantum $\mathfrak {sl}_{n}$ with an action of the current algebra $\mathbf {U}(\mathfrak {sl}_{n}[t])$ on its center.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:14 Jan 2016 10:51
Last Modified:08 Dec 2017 13:10
Publisher:National Center for Scientific Research, Institute of Mathematics
ISSN:0251-4184
Publisher DOI:https://doi.org/10.1007/s40306-014-0092-x

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