Vol. 279, No. 1-2, 2015

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ISSN: 0030-8730
Cellularity of certain quantum endomorphism algebras

Henning H. Andersen, Gustav I. Lehrer and Ruibin Zhang

Vol. 279 (2015), No. 1-2, 11–35

For any ring A˜ such that [q±12] A˜ (q12), let ΔA˜(d) be an A˜-form of the Weyl module of highest weight d of the quantised enveloping algebra UA˜ of sl2. For suitable A˜, we exhibit for all positive integers r an explicit cellular structure for EndU A˜(ΔA˜(d)r). This algebra and its cellular structure are described in terms of certain Temperley–Lieb-like diagrams. We also prove general results that relate endomorphism algebras of specialisations to specialisations of the endomorphism algebras. When ζ is a root of unity of order bigger than d we consider the Uζ-module structure of the specialisation Δζ(d)r at qζ of ΔA˜(d)r. As an application of these results, we prove that knowledge of the dimensions of the simple modules of the specialised cellular algebra above is equivalent to knowledge of the weight multiplicities of the tilting modules for Uζ(sl2). As an example, in the final section we independently recover the weight multiplicities of indecomposable tilting modules for Uζ(sl2) from the decomposition numbers of the endomorphism algebras, which are known through cellular theory.

quantum invariants, cellular algebras, tilting modules
Mathematical Subject Classification 2010
Primary: 17B37, 20G42
Secondary: 81R50
Received: 3 July 2014
Revised: 25 October 2014
Accepted: 25 October 2014
Published: 21 December 2015
Henning H. Andersen
QGM, Det Naturvidenskabelige Fakultet
Aarhus Universitet
Ny Munkegade Bygning 1530
DK-8000 Aarhus
Gustav I. Lehrer
School of Mathematics and Statistics F07
University of Sydney
NSW 2006
Sydney 2006
Ruibin Zhang
School of Mathematics and Statistics
University of Sydney
N.S.W. 2006
Sydney 2006