Finite-dimensional modules of the universal Askey–Wilson algebra and DAHA of type (C1∨,C1)

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Abstract

Assume that F is an algebraically closed field and let q denote a nonzero scalar in F that is not a root of unity. The universal Askey–Wilson algebra ▵q is a unital associative F-algebra defined by generators and relations. The generators are A, B, C and the relations state that each of A+qBC-q-1CBq2-q-2,B+qCA-q-1ACq2-q-2,C+qAB-q-1BAq2-q-2is central in ▵q. The universal DAHA (double affine Hecke algebra) Hq of type (C1∨,C1) is a unital associative F-algebra generated by {ti±1}i=03, and the relations state that titi-1=ti-1ti=1for alli=0,1,2,3;ti+ti-1is centralfor alli=0,1,2,3;t0t1t2t3=q-1.Each Hq-module is a ▵q-module by pulling back via the injection ▵q→ Hq given by A↦t1t0+(t1t0)-1,B↦t3t0+(t3t0)-1,C↦t2t0+(t2t0)-1.We classify the lattices of ▵q-submodules of finite-dimensional irreducible Hq-modules. As a corollary, for any finite-dimensional irreducible Hq-module V, the ▵q-module V is completely reducible if and only if t is diagonalizable on V.

Original languageEnglish
Article number81
JournalLetters in Mathematical Physics
Volume111
Issue number3
DOIs
StatePublished - Jun 2021

Keywords

  • Askey–Wilson algebras
  • Lattices
  • Representation theory

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