Notebook 2 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 double affine Hecke algebra H-tilde (Proposition 5.2) by generators and relations, it shows that, with (Subscript[T, 0]) defined by (Subscript[T, 1]^(-1))Y , the relations in Proposition 5.1 are also satisfied, and it shows that, with (Subscript[K, 0]), (Subscript[K, 1]) and (Subscript[T, 1]) given by (6.10), the relations for
tilde-AW(3, (Subscript[Q, 0])) (see (6.1)–(6.6)) are also satisfied.

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

Note an error in Proposition 5.1.
In formula (5.6) the factor q in the first term on the right should be (q^(-1)).

Last modified: September 18, 2015

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

<< NC`;

<<NCAlgebra`;

SNC[tt1,yy,yyinv,zz,zzinv];
SetCommutative[q,a,b,c,d];

rel1=yy**yyinv->1;
rel2=yyinv**yy->1;
rel3=zz**zzinv->1;
rel4=zzinv**zz->1;
rel5=tt1**tt1->-(a b+1) tt1-a b;
rel6=tt1**zz->zzinv**tt1+(a b+1) zzinv-a-b;
rel7=tt1**zzinv->zz**tt1-(a b+1) zzinv+a+b;
rel8=tt1**yy->(q^(-1))a b c d yyinv**tt1-(a b+1) yy+a b(1+(q^(-1))c d);
rel9=tt1**yyinv->q (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))  yy**tt1+q (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(a b+1) yy-(q (c^(-1)) (d^(-1))+1);
rel10=yy**zz->q zz**yy+(1+a b)c d zzinv**yyinv**tt1-(a+b)c d yyinv**tt1-(1+(q^(-1))c d)zzinv**tt1-(1-q)(1+a b)(1+(q^(-1))c d)zzinv+(c+d)tt1+(1-q)(a+b)(1+(q^(-1))c d);
rel11=yyinv**zz->(q^(-1)) zz**yyinv-q (a^(-1)) (b^(-1))(1+a b)  zzinv**yyinv**tt1+(a^(-1)) (b^(-1))(a+b) yyinv**tt1+q (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(1+(q^(-1))c d)zzinv**tt1+q (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(1-q) (1+a b) (1+(q^(-1))c d)zzinv-(a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(c+d)tt1-(a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(1-q)(1+a b) (c+d) ;
rel12=yy**zzinv->(q^(-1)) zzinv**yy-(q^(-2))c d(1+a b) zzinv**yyinv**tt1+(q^(-2))(a+b) c d yyinv**tt1+(q^(-1))(1+(q^(-1))c d) zzinv**tt1-(q^(-1))(c+d) tt1;
rel13=yyinv**zzinv->q zzinv**yyinv+q (a^(-1)) (b^(-1))(1+a b)  zzinv**yyinv**tt1-(a^(-1)) (b^(-1))(a+b)yyinv**tt1-(q^2) (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(1+(q^(-1))c d)zzinv**tt1+q (a^(-1)) (b^(-1))  (c^(-1)) (d^(-1))(c+d)tt1;

myrules={rel1,rel2,rel3,rel4,rel5,rel6,rel7,rel8,rel9,rel10,rel11,rel12,rel13};

tt1inv=-(a^(-1))(b^(-1))tt1-(1+(a^(-1))(b^(-1)));
tt0=tt1inv**yy;
tt0inv=-q (c^(-1))(d^(-1))tt0-(1+q (c^(-1))(d^(-1)));
kk0=yy+(q^(-1)) a b c d yyinv;
kk1=zz+zzinv;
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));
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));

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[tt1**tt1inv-1];
Sub[%,myrules];
NCE[%]

NCE[tt1inv**tt1-1];
Sub[%,myrules];
NCE[%]

NCE[tt0**tt0inv-1];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[tt0inv**tt0-1];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[tt0**tt0-(-((q^(-1))c d+1) tt0-(q^(-1))c d)];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[tt0**zz-(q zzinv**tt0-((q^(-1))c d+1)zz+(c+d))];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[tt0**zzinv-((q^(-1)) zz**tt0+((q^(-2))c d+(q^(-1))) zz-(q^(-1))(c+d))];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[(q+(q^(-1)))kk1**kk0**kk1-kk1**kk1**kk0-kk0**kk1**kk1-( bb kk1+cc0 kk0+dd0+ee  kk1**(tt1+ a b)+ff0(tt1+ a b))];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[(q+(q^(-1)))kk0**kk1**kk0-kk0**kk0**kk1-kk1**kk0**kk0-(bb kk0+cc1  kk1+dd1+ee  kk0**(tt1+a b)+ff1 (tt1+a b))];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]

NCE[qqtilde-qq0];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%];
Sub[%,myrules];
NCE[%]