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 | References | Similar Articles | Additional Information
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