Notebook 1 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 defines the extended AW(3) algebra (Definition 6.1) by generators and relations and it shows that the Casimir operator (6.5) indeed commutes with the generators.

The notebook uses the package NCAlgebra 4.0.6, which can be downloaded from
http://math.ucsd.edu/~ncalg/

Last modified: September 18, 2015

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

<< NC`;

<<NCAlgebra`;

SNC[tt1,kk0,kk1];
SetCommutative[q,a,b,c,d];

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);
ee=-(q^(-2))(((1-q))^3)(c+d);
ff0=(q^(-3))(((1-q))^3)(1+q)(c d+q) ;
ff1=(q^(-3))(((1-q))^3) (1+q)(a+b) c d;
gg=-(q^(-4))(((1-q))^3) ((a+b) (c+d)(c d((q^2)+1)+q)-q(a b+1)(((c^2)+(d^2) )(q+1)-c d)+(c d+e4) ((q^2)+1)+(e2+e4-a b) (q^3));

rel1=tt1**tt1->-(a b+1) tt1-a b;
rel2=tt1**kk0->kk0**tt1;
rel3=tt1**kk1->kk1**tt1;
rel4=kk1**kk1**kk0->(q+(q^(-1)))kk1**kk0**kk1-kk0**kk1**kk1-( bb kk1+cc0 kk0+dd0+ee  kk1**(tt1+a b)+ff0(tt1+ a b));
rel5=kk1**kk0**kk0->(q+(q^(-1)))kk0**kk1**kk0-kk0**kk0**kk1-(bb kk0+cc1  kk1+dd1+ee  kk0**(tt1+a b)+ff1 (tt1+a b));

myrules={rel1,rel2,rel3,rel4,rel5};

qqtilde=kk1**kk0**kk1**kk0-((q^2)+1+(q^(-2)))kk0**kk1**kk0**kk1+(q+(q^(-1)))kk0**kk0**kk1**kk1+(q+(q^(-1)))(cc0 kk0**kk0+cc1  kk1**kk1)+(bb+ee(tt1+a b))**((q+1+(q^(-1)))kk0**kk1+kk1**kk0)+ (q+1+(q^(-1)))(dd0+ff0(tt1+a b))**kk0+ (q+1+(q^(-1)))(dd1+ff1 (tt1+a b))**kk1+gg(tt1+a b);

NCE[qqtilde**kk0-kk0**qqtilde];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[qqtilde**kk1-kk1**qqtilde];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]