FRT presentation of classical Askey–Wilson algebras

Abstract

Automorphisms of the infinite-dimensional Onsager algebra are introduced. Certain quotients of the Onsager algebra are formulated using a polynomial in these automorphisms. In the simplest case, the quotient coincides with the classical analog of the Askey–Wilson algebra. In the general case, generalizations of the classical Askey–Wilson algebra are obtained. The corresponding class of solutions of the non-standard classical Yang–Baxter algebra is constructed, from which a generating function of elements in the commutative subalgebra is derived. We provide also another presentation of the Onsager algebra and of the classical Askey–Wilson algebras.

Appendix A

From (4.9) to (4.11), for $k=0,1,2$ one has:

$$\begin{aligned} A_0= & {} {\mathcal {W}}_0{,}\quad A_1={\mathcal {W}}_1{,}\quad G_1=-\frac{1}{4}\tilde{{\mathcal {G}}}_{1}{,}\\ A_{-1}= & {} 2{\mathcal {W}}_{-1}-{\mathcal {W}}_1{,}\quad A_{2}=2{\mathcal {W}}_{2}-{\mathcal {W}}_0{,}\quad G_2=-\frac{1}{2}\tilde{{\mathcal {G}}}_{2}{,}\\ A_{-2}= & {} 4{\mathcal {W}}_{-2}-{\mathcal {W}}_0-2{\mathcal {W}}_{2}{,}\quad A_{3}=4{\mathcal {W}}_{3}-{\mathcal {W}}_1-2{\mathcal {W}}_{-1}{,}\quad G_3=-\tilde{{\mathcal {G}}}_{3} + \frac{1}{4}\tilde{{\mathcal {G}}}_{1} {.} \end{aligned}$$

Conversely, from (4.12)–(4.13) for $k=1,2$ one has:

$$\begin{aligned} {\mathcal {W}}_{-1}= & {} \frac{A_1 + A_{-1}}{2}{,} \quad {\mathcal {W}}_{2}= \frac{A_0 + A_{2}}{2}{,}\quad \tilde{{\mathcal {G}}}_{2}=-2G_2{,} \\ {\mathcal {W}}_{-2}= & {} \frac{A_2 + 2A_{0} + A_{-2}}{4}{,} \quad {\mathcal {W}}_{2}= \frac{A_3 + 2A_{1} + A_{-1}}{4},\quad \tilde{{\mathcal {G}}}_{3}=-G_3 -2G_1\ . \end{aligned}$$