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Cellularity of certain quantum endomorphism algebras

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  • Henning Haahr Andersen
  • ,
  • Gus Lehrer, University of Sydney, Australia
  • Ruibin Zhang, University of Sydney, Australia
For any ring A˜ such that Z[q±1∕2]⊆A˜⊆Q(q1∕2), let ΔA˜(d) be an A˜-form of the Weyl module of highest weight d∈N of the quantised enveloping algebra UA˜ of sl2. For suitable A˜, we exhibit for all positive integers r an explicit cellular structure for EndUA˜(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 q↦ζ 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.
Original languageEnglish
JournalPacific Journal of Mathematics
Pages (from-to)11-35
Number of pages25
Publication statusPublished - 23 Dec 2015

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