Notebook 4 in connection with the paper
"The relationship between Zhedanov's algebra AW(3)
and the double affine Hecke algebra in the rank one case",
SIGMA 3 (2007), 063, 15 pages; arXiv:math.QA/0612730v4,
by Tom H. Koornwinder, T.H.Koornwinder@uva.nl

This checks that the algebra AW(3,(Subscript[Q, 0])), defined by (2.1)–(2.4), (2.7), (2.8) has a representation given by (2.5), (2.6).

Last modified: February 6, 2014

Off[General::spell, General::spell1]

v[z_]:=(((1-a z)(1-b z)(1-c z)(1-d z))/((1-(z^2))(1-q (z^2))))

ddsym[f_,z_]:=v[z]f[q z]+(1+(q^(-1))a b c d-v[z]-v[(z^(-1))])f[z]+v[(z^(-1))]f[(q^(-1))z]

zzsym[f_,z_]:=(z+(1/z))f[z]

e1=a+b+c+d;
e2=a b +a c+a d+b c+b d+c d;
e3=a b c+a b d+a c d+b c d;
e4=a b c d;
bb=(((1-(q^(-1))))^2) (e3+q e1);
cc0=(((q-(q^(-1))))^2);
cc1=(q^(-1))(((q-(q^(-1))))^2)e4;
dd0=-(1/(q^3))(((1-q))^2)  (1+q) (e4+q e2+(q^2));
dd1=-(q^(-3))(((1-q))^2) (1+q) (e1 e4+q e3);

(q+(q^(-1)))zzsym[ddsym[zzsym[f,#]&,#]&,z]-zzsym[zzsym[ddsym[f,#]&,#]&,z]-ddsym[zzsym[zzsym[f,#]&,#]&,z]-(bb zzsym[f,z]+cc0 ddsym[f,z]+dd0 f[z]);
Collect[%,{f[(q^(-1))z],f[z],f[q z]}];
Map[Simplify,%]

(q+(q^(-1)))ddsym[zzsym[ddsym[f,#]&,#]&,z]-ddsym[ddsym[zzsym[f,#]&,#]&,z]-zzsym[ddsym[ddsym[f,#]&,#]&,z]-(bb ddsym[f,z]+cc1 zzsym[f,z]+dd1 f[z]);
Collect[%,{f[(q^(-2))z],f[(q^(-1))z],f[z],f[q z],f[(q^2) z]}];
Map[Simplify,%]

qq1=((q^(-1/2))-(q^(3/2)))kk0**kk1**kk2+q kk2**kk2+bb(kk0**kk1+kk1**kk0)+cc0 q kk0**kk0+cc1 (q^(-1))kk1**kk1+dd0(1+q)kk0+dd1(1+(q^(-1)))kk1;

cas=zzsym[ddsym[zzsym[ddsym[f,#]&,#]&,#]&,z]-((q^2)+1+(q^(-2)))ddsym[zzsym[ddsym[zzsym[f,#]&,#]&,#]&,z]+(q+(q^(-1)))ddsym[ddsym[zzsym[zzsym[f,#]&,#]&,#]&,z]+(q+(q^(-1)))(cc0 ddsym[ddsym[f,#]&,z]+cc1 zzsym[zzsym[f,#]&,z])+bb((q+1+(q^(-1)))ddsym[zzsym[f,#]&,z]+ zzsym[ddsym[f,#]&,z])+(q+1+(q^(-1)))(dd0 ddsym[f,z]+dd1 zzsym[f,z]);
qq0=(q^(-4))(((1-q))^2) ((q^4)(e4-e2)+ (q^3)((e1^2)-e1 e3-2e2)-(q^2)(e2 e4+2 e4+e2)+q((e3^2)-2e2 e4-e1 e3)+e4(1-e2));
Collect[cas-qq0 f[z],{f[(q^(-2))z],f[(q^(-1))z],f[z],f[q z],f[(q^2) z]}];
Map[Simplify,%]