The Universal Askey – Wilson Algebra and DAHA of Type ( C ∨ 1 , C 1 )

Let F denote a field, and fix a nonzero q ∈ F such that q 6= 1. The universal Askey–Wilson algebra ∆q is the associative F-algebra defined by generators and relations in the following way. The generators are A, B, C. The relations assert that each of A+ qBC − q−1CB q2 − q−2 , B + qCA− q−1AC q2 − q−2 , C + qAB − q−1BA q2 − q−2 is central in ∆q. The universal DAHA Ĥq of type (C ∨ 1 , C1) is the associative F-algebra defined by generators {t±1 i }i=0 and relations (i) tit −1 i = t −1 i ti = 1; (ii) ti + t −1 i is central; (iii) t0t1t2t3 = q −1. We display an injection of F-algebras ψ : ∆q → Ĥq that sends A 7→ t1t0 + (t1t0), B 7→ t3t0 + (t3t0), C 7→ t2t0 + (t2t0). For the map ψ we compute the image of the three central elements mentioned above. The algebra ∆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 ∆q and Ĥq as a group of automorphisms. We show that ψ commutes with these B3 actions. Some related results are obtained.