Schur-Weyl duality for certain infinite dimensional \(\rm{U}_q(\mathfrak{sl}_2)\)-modules

Let \(V\) be the two-dimensional simple module and \(M\) be a projective Verma module for the quantum group of \(\mathfrak{sl}_2\) at generic \(q\). We show that for any \(r\ge 1\), the endomorphism algebra of \(M\otimes V^{\otimes r}\) is isomorphic to the type \(B\) Temperley-Lieb algebra \(\rm{TL...

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Bibliographic Details
Published in:arXiv.org 2019-01
Main Authors: Iohara, Kenji, Lehrer, Gus, Zhang, Ruibin
Format: Article
Language:English
Subjects:
Algebra
Cellular structure
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Summary:Let \(V\) be the two-dimensional simple module and \(M\) be a projective Verma module for the quantum group of \(\mathfrak{sl}_2\) at generic \(q\). We show that for any \(r\ge 1\), the endomorphism algebra of \(M\otimes V^{\otimes r}\) is isomorphic to the type \(B\) Temperley-Lieb algebra \(\rm{TLB}_r(q, Q)\) for an appropriate parameter \(Q\) depending on \(M\). The parameter \(Q\) is determined explicitly. We also use the cellular structure to determine precisely for which values of \(r\) the endomorphism algebra is semisimple. A key element of our method is to identify the algebras \(\rm{TLB}_r(q,Q)\) as the endomorphism algebras of the objects in a quotient category of the category of coloured ribbon graphs of Freyd-Yetter or the tangle diagrams of Turaev and Reshitikhin.
ISSN:2331-8422