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Equivalence of a tangle category and a category of infinite dimensional $\mathrm {U}_q(\mathfrak {sl}_2)$-modules


Authors: K. Iohara, G. I. Lehrer and R. B. Zhang
Journal: Represent. Theory 25 (2021), 265-299
MSC (2020): Primary 17B37, 20G42; Secondary 81R50
DOI: https://doi.org/10.1090/ert/568
Published electronically: April 20, 2021
MathSciNet review: 4245314
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Abstract: It is very well known that if $V$ is the simple $2$-dimensional representation of $\mathrm {U}_q(\mathfrak {sl}_2)$, the category of representations $V^{\otimes r}$, $r=0,1,2,\dots$, is equivalent to the Temperley-Lieb category $\mathrm {TL}(q)$. Such categorical equivalences between tangle categories and categories of representations are rare. In this work we give a family of new equivalences by extending the above equivalence to one between the category of representations $M\otimes V^{\otimes r}$, where $M$ is a projective Verma module of $\mathrm {U}_q(\mathfrak {sl}_2)$ and the type $B$ Temperley-Lieb category $\mathbb {TLB}(q,Q)$, realised as a subquotient of the tangle category of Freyd, Yetter, Reshetikhin, Turaev and others.


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Additional Information

K. Iohara
Affiliation: Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne cedex, France
MR Author ID: 334495
ORCID: 0000-0001-6748-8256
Email: iohara@math.univ-lyon1.fr

G. I. Lehrer
Affiliation: School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
MR Author ID: 112045
ORCID: 0000-0002-7918-7650
Email: gustav.lehrer@sydney.edu.au

R. B. Zhang
Affiliation: School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
Email: ruibin.zhang@sydney.edu.au

Keywords: Tangle category, Temperley-Lieb category of type $B$, Verma module.
Received by editor(s): November 20, 2019
Received by editor(s) in revised form: November 25, 2020
Published electronically: April 20, 2021
Additional Notes: Partially supported by the Australian Research Council.
Article copyright: © Copyright 2021 American Mathematical Society