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

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- by K. Iohara, G. I. Lehrer and R. B. Zhang
- Represent. Theory
**25**(2021), 265-299 - DOI: https://doi.org/10.1090/ert/568
- Published electronically: April 20, 2021
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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.## References

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## Bibliographic 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
- 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.
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 265-299 - MSC (2020): Primary 17B37, 20G42; Secondary 81R50
- DOI: https://doi.org/10.1090/ert/568
- MathSciNet review: 4245314