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Trace as an Alternative Decategorification Functor

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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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Acknowledgments

A.B. and Z.G. were supported by Swiss National Science Foundation under Grant PDFMP2-141752/1. K.H. was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 24540077. A.D.L. was partially supported by NSF grant DMS-1255334 and by the John Templeton Foundation. A.D.L. is extremely grateful to Mikhail Khovanov for sharing his insights and vision about higher representation theory. Some of the ideas and calculations appearing in this article were done as part of this collaboration. He is also grateful to Arun Ram for helpful decategorification discussions. A.B. would like to thank Benjamin Cooper for helpful conversations.

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Authors and Affiliations

  1. Universität Zürich, Winterthurerstr. 190, CH-8057, Zürich, Switzerland

    Anna Beliakova & Zaur Guliyev

  2. Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

    Kazuo Habiro

  3. University of Southern California, Los Angeles, CA, 90089, USA

    Aaron D. Lauda

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  1. Anna Beliakova
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  2. Zaur Guliyev
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Correspondence to Anna Beliakova.

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Beliakova, A., Guliyev, Z., Habiro, K. et al. Trace as an Alternative Decategorification Functor. Acta Math Vietnam 39, 425–480 (2014). https://doi.org/10.1007/s40306-014-0092-x

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