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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) H_{q} 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 H_{q}-module is a ▵_{q}-module by pulling back via the injection ▵_{q}→ H_{q} 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 H_{q}-modules. As a corollary, for any finite-dimensional irreducible H_{q}-module V, the ▵_{q}-module V is completely reducible if and only if t is diagonalizable on V.

Original language | English |
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Article number | 81 |

Journal | Letters in Mathematical Physics |

Volume | 111 |

Issue number | 3 |

DOIs | |

State | Published - Jun 2021 |

## Keywords

- Askey–Wilson algebras
- Lattices
- Representation theory

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