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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 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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Dive into the research topics of 'Finite-dimensional modules of the universal Askey–Wilson algebra and DAHA of type (C1∨,C1)'. Together they form a unique fingerprint.Projects
- 1 Finished
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The Unversial Racah Algebra and Its Applications(4/4)
Huang, H.-W. (PI)
1/08/20 → 31/07/21
Project: Research