The Universal Askey-Wilson Algebra and DAHA of Type (C₁^\vee,C₁)

2012/02/29 by Paul Terwilliger · 1 citation
Mathematics · #math.QA #msc:33D80

paper · pdf · doi:10.3842/sigma.2013.047

published as SIGMA 9 (2013), 047, 40 pages

arxiv created 2013/07/15 · arxiv updated 2013/07/16

Abstract

Let \mathbb F denote a field, and fix a nonzero q∈\mathbb F such that q⁴\not=1. The universal Askey-Wilson algebra Δ_q is the associative \mathbb F-algebra defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A+\fracqBC-q^-1CBq²-q^-2, B+\fracqCA-q^-1ACq²-q^-2, C+\fracqAB-q^-1BAq²-q^-2 is central in Δ_q. The universal DAHA H_q of type (C₁^\vee,C₁) is the associative \mathbb F-algebra defined by generators \lbrace t^±1_i\rbrace_i=0³ and relations (i) t_i t^-1_i=t^-1_i t_i=1; (ii) t_i+t^-1_i is central; (iii) t₀t₁t₂t₃=q^-1. We display an injection of \mathbb F-algebras ψ:Δ_q→ H_q that sends A↦ t₁t₀+(t₁t₀)^-1, B↦ t₃t₀+(t₃t₀)^-1, C↦ t₂t₀+(t₂t₀)^-1. For the map ψ we compute the image of the three central elements mentioned above. The algebra Δ_q has another central element of interest, called the Casimir element Ω. We compute the image of Ω under ψ. We describe how the Artin braid group B₃ acts on Δ_q and H_q as a group of automorphisms. We show that ψ commutes with these B₃ actions. Some related results are obtained.

Cited by

• Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

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• Double Affine Hecke Algebras of Rank 1 and the Z₃-Symmetric Askey-Wilson Relations
• A higher rank extension of the Askey-Wilson Algebra
• Finite-dimensional irreducible modules of the universal Askey--Wilson algebra at roots of unity
• The Universal Askey-Wilson Algebra and the Equitable Presentation of U q (sl 2 )