R:= RootSystem( "A",7);
 U:=QuantizedUEA(R); 
 g:= GeneratorsOfAlgebra( U ); 
 PositiveRoots(R); 
 PositiveRootsInConvexOrder(R); 
########## generators of U ###############
 f1:=g[1]; f2:=g[3]; f3:=g[6]; f4:=g[10]; f5:=g[15]; f6:=g[21]; f7:=g[28];
 e1:=g[43]; e2:=g[45]; e3:=g[48]; e4:=g[52]; e5:=g[57]; e6:=g[63]; e7:= g[70];
 k1:=g[29]; k2:=g[31]; k3:=g[33]; k4:=g[35]; k5:=g[37]; k6:=g[39]; k7:=g[41];
 k1m:=g[30]; k2m:=g[32]; k3m:=g[34]; k4m:=g[36]; k5m:=g[38]; k6m:=g[40]; k7m:=g[42];   
########## generators of B ###############
 b1:=f1 - k1m * e7;
 b2:=f2 - k2m * e6;
 b3:=f3 - k3m * e5;
 b4:=f4 - k4m * e4;
 b5:=f5 - k5m * e3;
 b6:=f6 - k6m * e2;
 b7:=f7 - k7m * e1;
 a1:= k1 * k7m;
 a2:= k2 * k6m;
 a3:= k3 * k5m;
 a5:= k5 * k3m;
 a6:= k6 * k2m;
 a7:= k7 * k1m;
########## q-numbers #################### 
 2q:=_q+_q^-1;
########## q-commutator #################
 qcom2:=function(a,b)
   return a*b-_q*b*a;
 end;
######### relations in B as a function #########
 rel1:=function(a,b)
   return a^2*b - (_q + _q^-1)*a*b*a + b*a^2 ;
 end;
################# relations ####################
(_q-_q^-1)*(b1 * b7 - b7 * b1) - a1 + a7;
(_q-_q^-1)*(b2 * b6 - b6 * b2) - a2 + a6;
(_q-_q^-1)*(b3 * b5 - b5 * b3) - a3 + a5;
rel1(b1,b2);
rel1(b2,b1);
b1 *b3 - b3* b1;
b2 * b4 - b4 * b2;
rel1(b2,b3);
rel1(b3,b2);
rel1(b3,b4);
rel1(b5,b4);
rel1(b4,b3) + _q^-1 * b3;
rel1(b4,b5) + _q^-1 * b5;  #### from here onwards it's symmetric
rel1(b5,b6);
rel1(b6,b5);
################# \tau_4^{-1} Braid group action #####################
t4mb1:=b1;;
t4mb2:=b2;;
t4mb3:=qcom2(b4,b3);;
t4mb4:=b4;;
t4mb5:=qcom2(b4,b5);;
t4mb6:=b6;;
t4mb7:=b7;;
t4ma1:=a1;;
t4ma2:=a2;;
t4ma3:=a3;;
t4ma5:=a5;;
t4ma6:=a6;;
t4ma7:=a7;;
################# t4m is an algebra homomorphism ############
(_q-_q^-1)*(t4mb3 * t4mb5 - t4mb5 * t4mb3) - t4ma3 + t4ma5;
rel1(t4mb2,t4mb3);
rel1(t4mb3,t4mb2);
rel1(t4mb3, t4mb4);
rel1(t4mb5, t4mb4);
rel1(t4mb4, t4mb3) + _q^-1 * t4mb3;
rel1(t4mb4, t4mb5) + _q^-1 * t4mb5;
rel1(t4mb5,t4mb6);
rel1(t4mb6,t4mb5);
################# the inverse \tau_4 of \tau_4^{-1} ##################
t4b1:=b1;;
t4b2:=b2;;
t4b3:=qcom2(b3,b4);;
t4b4:=b4;;
t4b5:=qcom2(b5,b4);;
t4b6:=b6;;
t4b7:=b7;;
t4a1:=a1;;
t4a2:=a2;;
t4a3:=a3;;
t4a5:=a5;;
t4a6:=a6;;
t4a7:=a7;;
################ t4 is an algebra homomorphism ################
(_q-_q^-1)*(t4b3 * t4b5 - t4b5 * t4b3) - t4a3 + t4a5;
rel1(t4b2,t4b3);
rel1(t4b3,t4b2);
rel1(t4b3,t4b4);
rel1(t4b5,t4b4);
rel1(t4b4,t4b3) + _q^-1 * t4b3;
rel1(t4b4,t4b5) + _q^-1 * t4b5;
rel1(t4b5,t4b6);
rel1(t4b6,t4b5);
########### t4 is the inverse of t4m ####################
qcom2(t4mb3,t4mb4)-b3;
qcom2(t4mb5,t4mb4)-b5;
############ \tau_3^- braid group action ####################
t3mb1:=b1;;
t3mb2:= qcom2(b3,b2);; #### times q^(-1/2)
t3mb3:= _q^-1 * a3* b5;;
t3mb4:= _q^-1 * qcom2(b3,qcom2(b5,b4))+b4*a3;;
t3mb5:= _q^-1 * a5 * b3;;
t3mb6:=  qcom2(b5,b6);;   #### times _q^(-1/2)
t3mb7:= b7;;
t3ma1:= a1;;
t3ma2:= a2*a3;;
t3ma3:= a5;;
t3ma5:= a3;;
t3ma6:= a6*a5;;
t3ma7:= a7;;
############ t3m is an algebra homomorphism #############
(_q-_q^-1)*(t3mb1 * t3mb7 - t3mb7 * t3mb1) - t3ma1 + t3ma7;
(_q-_q^-1)* _q^-1* (t3mb2 * t3mb6 - t3mb6 * t3mb2) - t3ma2 + t3ma6;
(_q-_q^-1)*(t3mb3 * t3mb5 - t3mb5 * t3mb3) - t3ma3 + t3ma5;
rel1(t3mb1,t3mb2);
rel1(t3mb2,t3mb1);
t3mb1 * t3mb3 - t3mb3 * t3mb1;
t3mb2 * t3mb4 - t3mb4 * t3mb2;
rel1(t3mb2,t3mb3);
rel1(t3mb3,t3mb2);
rel1(t3mb3,t3mb4);
rel1(t3mb5,t3mb4);
rel1(t3mb4,t3mb3) + _q^-1 * t3mb3;
########### the inverse \tau_3 of \tau_3^- #################
t3b1:= b1;;
t3b2:= qcom2(b2,b3);; #### times _q^(-1/2)
t3b3:= _q * a5* b5;;
t3b4:= _q^-1 * qcom2(qcom2(b4,b3),b5)+ b4*a3;;
t3b5:= _q * a3 * b3;;
t3b6:= qcom2(b6,b5);; #### times _q^(-1/2)
t3b7:= b7;;
t3a1:= a1;;
t3a2:= a2*a3;;
t3a3:= a5;;
t3a5:= a3;;
t3a6:= a6*a5;;
t3a7:= a7;;
########### t3 is an algebra homomorphism ############
(_q-_q^-1)*(t3b1 * t3b7 - t3b7 * t3b1) - t3a1 + t3a7;
(_q-_q^-1)* _q^-1 * (t3b2 * t3b6 - t3b6 * t3b2) - t3a2 + t3a6;
(_q-_q^-1)*(t3b3 * t3b5 - t3b5 * t3b3) - t3a3 + t3a5;
rel1(t3b1,t3b2);
rel1(t3b2,t3b1);
t3b1 * t3b3 - t3b3 * t3b1;
t3b2 * t3b4 - t3b4 * t3b2;
rel1(t3b2,t3b3);
rel1(t3b3,t3b2);
rel1(t3b3,t3b4);
rel1(t3b5,t3b4);
rel1(t3b4,t3b3) + _q^-1 * t3b3;
########### t3 is the inverse of t3m ##############
_q^-1 * qcom2(t3b3,t3b2)-b2;
_q^-1 * t3a3* t3b5 - b3;
_q^-1 * qcom2(t3b3,qcom2(t3b5,t3b4))+t3b4*t3a3 - b4;
_q^-1 * t3a5 * t3b3 - b5;
_q^-1 * qcom2(t3b5,t3b6) - b6;
 t3a2*t3a3 - a2;
t3a5 - a3;
t3a3 - a5;
 t3a6*t3a5 - a6;
############ \tau_2^- braid group action ############
t2mb1:=qcom2(b2,b1);; #### times _q^(-1/2)
t2mb2:= _q^-1*a2 * b6;;
t2mb3:=qcom2(b2,b3);; #### times _q^(-1/2)
t2mb4:=b4;;
t2mb5:=qcom2(b6,b5);; #### times _q^(-1/2)
t2mb6:= _q^-1 * a6 *b2;;
t2mb7:=qcom2(b6,b7);; #### times _q^(-1/2)
t2ma1:= a1 * a2;;
t2ma2:= a6;;
t2ma3:= a2 * a3;;
t2ma5:= a5 * a6;;
t2ma6:= a2;;
t2ma7:= a6 * a7;;
################# t2m is an algebra homomorphism ####################
(_q-_q^-1)* _q^-1 * (t2mb1 * t2mb7 - t2mb7 * t2mb1) - t2ma1 + t2ma7;
(_q-_q^-1)*(t2mb2 * t2mb6 - t2mb6 * t2mb2) - t2ma2 + t2ma6;
(_q-_q^-1)* _q^-1 * (t2mb3 * t2mb5 - t2mb5 * t2mb3) - t2ma3 + t2ma5;
rel1(t2mb1,t2mb2);
rel1(t2mb2,t2mb1);
t2mb1 * t2mb3 - t2mb3 * t2mb1;
t2mb2 * t2mb4 - t2mb4 * t2mb2;
rel1(t2mb2,t2mb3);
rel1(t2mb3,t2mb2);
rel1(t2mb3,t2mb4);
rel1(t2mb5,t2mb4);
rel1(t2mb4,t2mb3) + _q^-1 * t2mb3;
rel1(t2mb4,t2mb5) + _q^-1 * t2mb5;  #### from here onwards it's symmetric
################ the inverse \tau_2 of \tau_2^- #####################
t2b1:= qcom2(b1,b2);;  #### times _q^(-1/2)
t2b2:= _q *a6 * b6;;
t2b3:= qcom2(b3,b2);;  #### times _q^(-1/2)
t2b4:= b4;;
t2b5:= qcom2(b5,b6);;  #### times _q^(-1/2)
t2b6:= _q * a2 *b2;;
t2b7:= qcom2(b7,b6);;  #### times _q^(-1/2)
t2a1:= a1 * a2;;
t2a2:= a6;;
t2a3:= a2 * a3;;
t2a5:= a5 * a6;;
t2a6:= a2;;
t2a7:= a6 * a7;;
################# t2 is an algebra homomorphism ####################
(_q-_q^-1)* _q^-1 * (t2b1 * t2b7 - t2b7 * t2b1) - t2a1 + t2a7;
(_q-_q^-1)*(t2b2 * t2b6 - t2b6 * t2b2) - t2a2 + t2a6;
(_q-_q^-1)* _q^-1 * (t2b3 * t2b5 - t2b5 * t2b3) - t2a3 + t2a5;
rel1(t2b1,t2b2);
rel1(t2b2,t2b1);
t2b1 * t2b3 - t2b3 * t2b1;
t2b2 * t2b4 - t2b4 * t2b2;
rel1(t2b2,t2b3);
rel1(t2b3,t2b2);
rel1(t2b3,t2b4);
rel1(t2b5,t2b4);
rel1(t2b4,t2b3) + _q^-1 * t2b3;
rel1(t2b4,t2b5) + _q^-1 * t2b5;  #### from here onwards it's symmetric
############### t2 is the inverse of t2m ####################
_q^-1 * qcom2(t2b2,t2b1) -b1;
_q^-1 *t2a2 * t2b6 -b2;
_q^-1 * qcom2(t2b2,t2b3) -b3;
t2b4 -b4;
_q^-1 * qcom2(t2b6,t2b5) -b5;
_q^-1 * t2a6 *t2b2 -b6;
_q^-1 * qcom2(t2b6,t2b7) -b7;
t2a1 * t2a2 - a1;
t2a6 -a2;
t2a2 * t2a3 -a3;
t2a5 * t2a6 -a5;
t2a2 -a6;
t2a6 * t2a7 -a7;
############ \tau_1^- braid group action ############
t1mb1:= _q^-1 * a1 * b7;;
t1mb2:= qcom2(b1,b2);; #### times _q^(-1/2)
t1mb3:=b3;;
t1mb4:=b4;;
t1mb5:=b5;;
t1mb6:= qcom2(b7,b6);; #### times _q^(-1/2)
t1mb7:= _q^-1 * a7 * b1;;
t1ma1:= a7;;
t1ma2:= a1 *a2;;
t1ma3:= a3;;
t1ma5:= a5;;
t1ma6:= a7* a6;;
t1ma7:= a1;;
################# t1m is an algebra homomorphism ####################
(_q-_q^-1)*(t1mb1 * t1mb7 - t1mb7 * t1mb1) - t1ma1 + t1ma7;
(_q-_q^-1)* _q^-1 * (t1mb2 * t1mb6 - t1mb6 * t1mb2) - t1ma2 + t1ma6;
(_q-_q^-1)*(t1mb3 * t1mb5 - t1mb5 * t1mb3) - t1ma3 + t1ma5;
rel1(t1mb1,t1mb2);
rel1(t1mb2,t1mb1);
t1mb1 * t1mb3 - t1mb3 * t1mb1;
t1mb2 * t1mb4 - t1mb4 * t1mb2;
rel1(t1mb2,t1mb3);
rel1(t1mb3,t1mb2);
rel1(t1mb3,t1mb4);
rel1(t1mb5,t1mb4);
rel1(t1mb4,t1mb3) + _q^-1 * t1mb3;
rel1(t1mb4,t1mb5) + _q^-1 * t1mb5;  #### from here onwards it's symmetric
########### the inverse \tau_1 of \tau_1^- ############
t1b1:= _q * a7 * b7;;
t1b2:= qcom2(b2,b1);; #### times _q^(-1/2)
t1b3:=b3;;
t1b4:=b4;;
t1b5:=b5;;
t1b6:= qcom2(b6,b7);; #### times _q^(-1/2)
t1b7:= _q * a1 * b1;;
t1a1:= a7;;
t1a2:= a1 *a2;;
t1a3:= a3;;
t1a5:= a5;;
t1a6:= a7* a6;;
t1a7:= a1;;
################# t1 is an algebra homomorphism ####################
(_q-_q^-1)*(t1b1 * t1b7 - t1b7 * t1b1) - t1a1 + t1a7;
(_q-_q^-1)* _q^-1 * (t1b2 * t1b6 - t1b6 * t1b2) - t1a2 + t1a6;
(_q-_q^-1)*(t1b3 * t1b5 - t1b5 * t1b3) - t1a3 + t1a5;
rel1(t1b1,t1b2);
rel1(t1b2,t1b1);
t1b1 * t1b3 - t1b3 * t1b1;
t1b2 * t1b4 - t1b4 * t1b2;
rel1(t1b2,t1b3);
rel1(t1b3,t1b2);
rel1(t1b3,t1b4);
rel1(t1b5,t1b4);
rel1(t1b4,t1b3) + _q^-1 * t1b3;
rel1(t1b4,t1b5) + _q^-1 * t1b5;  #### from here onwards it's symmetric
############ t1m is the iverse of t1 ####################
_q^-1 * t1a1 * t1b7 -b1;
_q^-1 * qcom2(t1b1,t1b2) -b2;
t1b3 -b3;
t1b4 -b4;
t1b5 -b5;
_q^-1 * qcom2(t1b7,t1b6) -b6;
_q^-1 * t1a7 * t1b1 -b7;
t1a7 -a1;
 t1a1 *t1a2 -a2;
t1a3 -a3;
t1a5 -a5;
 t1a7* t1a6 -a6;
t1a1 -a7;
############ braid relations for t4 and t3 ###############################
##### To simplify notation we write t434b5 instead of t4 t3 t4 (b5) ######
##### And similar notation for other evaluations of products of t4   ######
##### and t3 or other braid group generators                        ######
###########################################################################
########### recall the definition of t3: #################################
t3b1:=b1;;
t3b2:= qcom2(b2,b3);; #### times _q^(-1/2)
t3b3:=_q * a5* b5;;
t3b4:=_q^-1 * qcom2(qcom2(b4,b3),b5)+ b4*a3;;
t3b5:=_q * a3 * b3;;
t3b6:= qcom2(b6,b5);; #### times _q^(-1/2)
t3b7:=b7;;
t3a1:= a1;;
t3a2:= a2*a3;;
t3a3:= a5;;
t3a5:= a3;;
t3a6:= a6*a5;;
t3a7:= a7;;
########### and recall the definition of t4 ###############################
t4b1:=b1;;
t4b2:=b2;;
t4b3:=qcom2(b3,b4);;
t4b4:=b4;;
t4b5:=qcom2(b5,b4);;
t4b6:=b6;;
t4b7:=b7;;
t4a1:=a1;;
t4a2:=a2;;
t4a3:=a3;;
t4a5:=a5;;
t4a6:=a6;;
t4a7:=a7;;
##############################################################################
#### Observe, that both t3 and t4 act as the identity on  b1, a1, b7, a7 ####
#### Hence we only need to consider the action on the smaller subalgebra  ####
#### generated by the other generators                                    ####
##############################################################################
t43b2:= qcom2(t4b2,t4b3);; #### times _q^(-1/2)
t43b3:=_q * t4a5* t4b5;;
t43b4:=_q^-1 * qcom2(qcom2(t4b4,t4b3),t4b5)+ t4b4*t4a3;;
t43b5:=_q * t4a3 * t4b3;;
t43b6:= qcom2(t4b6,t4b5);; #### times _q^(-1/2)
t43a2:= t4a2*t4a3;;
t43a3:= t4a5;;
t43a5:= t4a3;;
t43a6:= t4a6*t4a5;;
#############
t34b2:=t3b2;; #### times _q^(-1/2)
t34b3:=qcom2(t3b3,t3b4);;
t34b4:=t3b4;;
t34b5:=qcom2(t3b5,t3b4);;
t34b6:=t3b6;; #### times _q^(-1/2)
t34a2:=t3a2;;
t34a3:=t3a3;;
t34a5:=t3a5;;
t34a6:=t3a6;;
############
t343b2:= qcom2(t34b2,t34b3);; #### times _q^-1
t343b3:=_q * t34a5* t34b5;;
t343b4:=_q^-1 * qcom2(qcom2(t34b4,t34b3),t34b5)+ t34b4*t34a3;;
t343b5:=_q * t34a3 * t34b3;;
t343b6:= qcom2(t34b6,t34b5);; #### times _q^-1
t343a2:= t34a2*t34a3;;
t343a3:= t34a5;;
t343a5:= t34a3;;
t343a6:= t34a6*t34a5;;
#############
t434b2:=t43b2;; #### times _q^(-1/2)
t434b3:=qcom2(t43b3,t43b4);;
t434b4:=t43b4;;
t434b5:=qcom2(t43b5,t43b4);;
t434b6:=t43b6;; #### times _q^(-1/2)
t434a2:=t43a2;;
t434a3:=t43a3;;
t434a5:=t43a5;;
t434a6:=t43a6;;
############
t4343b2:=qcom2(t434b2,t434b3);; #### times _q^-1
t4343b3:=_q * t434a5* t434b5;;
t4343b4:=_q^-1 * qcom2(qcom2(t434b4,t434b3),t434b5)+ t434b4*t434a3;;
t4343b5:=_q * t434a3 * t434b3;;
t4343b6:=qcom2(t434b6,t434b5);; #### times _q^-1
t4343a2:= t434a2*t434a3;; 
t4343a3:= t434a5;;
t4343a5:= t434a3;;
t4343a6:= t434a6*t434a5;;
#############
t3434b2:=t343b2;; #### times _q^-1
t3434b3:=qcom2(t343b3,t343b4);;
t3434b4:=t343b4;;
t3434b5:=qcom2(t343b5,t343b4);;
t3434b6:=t343b6;; #### times _q^-1
t3434a2:=t343a2;;
t3434a3:=t343a3;;
t3434a5:=t343a5;;
t3434a6:=t343a6;;
####################################################################
############ final verification of the type B braid relation: ######
############ all the followsing computations need to give 0   ######
####################################################################
t3434b2 - t4343b2; ##### times q^-1
t3434b3 - t4343b3;
t3434b4 - t4343b4;
t3434b5 - t4343b5;
t3434b6 - t4343b6; ##### times q^-1
t3434a2 - t4343a2;
t3434a3 - t4343a3;
t3434a5 - t4343a5;
t3434a6 - t4343a6;
####################################################################
#### Now we approach the type A braid relation between t2 and #####
#### t3 in the same way, again writing t232b1 etc.            #####
####################################################################
t2b1:=qcom2(b1,b2);; #### times _q^(-1/2)
t2b2:= _q *a6 * b6;;
t2b3:=qcom2(b3,b2);; #### times _q^(-1/2)
t2b4:=b4;;
t2b5:=qcom2(b5,b6);; #### times _q^(-1/2)
t2b6:= _q * a2 *b2;;
t2b7:=qcom2(b7,b6);; #### times _q^(-1/2)
t2a1:= a1 * a2;;
t2a2:= a6;;
t2a3:= a2 * a3;;
t2a5:= a5 * a6;;
t2a6:= a2;;
t2a7:= a6 * a7;;
#########################
t3b1:=b1;;
t3b2:=qcom2(b2,b3);; #### times _q^(-1/2)
t3b3:=_q * a5* b5;;
t3b4:=_q^-1 * qcom2(qcom2(b4,b3),b5)+ b4*a3;;
t3b5:=_q * a3 * b3;;
t3b6:=qcom2(b6,b5);; #### times _q^(-1/2)
t3b7:=b7;;
t3a1:= a1;;
t3a2:= a2*a3;;
t3a3:= a5;;
t3a5:= a3;;
t3a6:= a6*a5;;
t3a7:= a7;;
#########################
t32b1:=qcom2(t3b1,t3b2);;  #### times _q^-1
t32b2:= _q * t3a6 * t3b6;; 
t32b3:=qcom2(t3b3,t3b2);;  #### times _q^-1
t32b4:=t3b4;;
t32b5:=qcom2(t3b5,t3b6);;  #### times _q^-1
t32b6:= _q * t3a2 *t3b2;;  
t32b7:=qcom2(t3b7,t3b6);;  #### times _q^-1
t32a1:= t3a1 * t3a2;;
t32a2:= t3a6;;
t32a3:= t3a2 * t3a3;;
t32a5:= t3a5 * t3a6;;
t32a6:= t3a2;;
t32a7:= t3a6 * t3a7;;
#########################
t23b1:=t2b1;;              #### times _q^(-1/2) 
t23b2:=qcom2(t2b2,t2b3);;  #### times _q^-1
t23b3:= _q * t2a5* t2b5;;  #### times _q^(-1/2)
t23b4:= _q^-2 * qcom2(qcom2(t2b4,t2b3),t2b5)+ t2b4*t2a3;; #### q^-1 entered only in first term
t23b5:=_q * t2a3 * t2b3;;  #### times _q^(-1/2)
t23b6:=qcom2(t2b6,t2b5);;  #### times _q^-1
t23b7:=t2b7;;              #### times _q^(-1/2) 
t23a1:= t2a1;;
t23a2:= t2a2*t2a3;;
t23a3:= t2a5;;
t23a5:= t2a3;;
t23a6:= t2a6*t2a5;;
t23a7:= t2a7;;
#########################
t232b1:= qcom2(t23b1,t23b2);; #### times _q^-2
t232b2:= _q *t23a6 * t23b6;;  #### times _q^(-1/2)
t232b3:= qcom2(t23b3,t23b2);; #### times _q^-2
t232b4:=t23b4;;
t232b5:= qcom2(t23b5,t23b6);; #### times _q^-2
t232b6:= _q * t23a2 *t23b2;;  #### times _q^(-1/2)
t232b7:= qcom2(t23b7,t23b6);; #### times _q^-2
t232a1:= t23a1 * t23a2;;
t232a2:= t23a6;;
t232a3:= t23a2 * t23a3;;
t232a5:= t23a5 * t23a6;;
t232a6:= t23a2;;
t232a7:= t23a6 * t23a7;;
#########################
t323b1:=t32b1;;               #### times _q^-1
t323b2:=qcom2(t32b2,t32b3);;  #### times _q^(-3/2)
t323b3:=_q * t32a5* t32b5;;   #### times _q^-1
t323b4:= _q^-3 * qcom2(qcom2(t32b4,t32b3),t32b5)+ t32b4*t32a3;; #### again _q^-1 only enters first term
t323b5:=_q * t32a3 * t32b3;;  #### times _q^-1
t323b6:=qcom2(t32b6,t32b5);;  #### times _q^(-3/2)
t323b7:=t32b7;;               #### times _q^-1
t323a1:= t32a1;;
t323a2:= t32a2*t32a3;;
t323a3:= t32a5;;
t323a5:= t32a3;;
t323a6:= t32a6*t32a5;;
t323a7:= t32a7;;
###############################################################################
########## Final step in the verification of the type A braid relation ########
########## We need to take into account the differences of q-factors   ########
###############################################################################
t323b1 - _q^-1 * t232b1;
t323b2 - _q * t232b2;
t323b3 - _q^-1 * t232b3;
t323b4 - t232b4;
t323b5 - _q^-1 * t232b5;
t323b6 - _q * t232b6;
t323b7 - _q^-1 * t232b7;
t323a1 - t232a1;
t323a2 - t232a2;
t323a3 - t232a3;
t323a5 - t232a5;
t323a6 - t232a6;
t323a7 - t232a7;
###############################################################################
########## We now verify the type A braid relation between t1 and t2 ########
###############################################################################
t1b1:= _q * a7 * b7;;
t1b2:= qcom2(b2,b1);; #### times _q^(-1/2)
t1b3:=b3;;
t1b4:=b4;;
t1b5:=b5;;
t1b6:= qcom2(b6,b7);; #### times _q^(-1/2)
t1b7:= _q * a1 * b1;;
t1a1:= a7;;
t1a2:= a1 *a2;;
t1a3:= a3;;
t1a5:= a5;;
t1a6:= a7* a6;;
t1a7:= a1;;
###################
t2b1:=qcom2(b1,b2);; #### times _q^(-1/2)
t2b2:= _q *a6 * b6;;
t2b3:=qcom2(b3,b2);; #### times _q^(-1/2)
t2b4:=b4;;
t2b5:=qcom2(b5,b6);; #### times _q^(-1/2)
t2b6:= _q * a2 *b2;;
t2b7:=qcom2(b7,b6);; #### times _q^(-1/2)
t2a1:= a1 * a2;;
t2a2:= a6;;
t2a3:= a2 * a3;;
t2a5:= a5 * a6;;
t2a6:= a2;;
t2a7:= a6 * a7;;
#####################
t21b1:= _q * t2a7 * t2b7;; #### times _q^(-1/2)
t21b2:= qcom2(t2b2,t2b1);; #### times _q^-1
t21b3:=t2b3;;              #### times _q^(-1/2)
t21b4:=t2b4;;
t21b5:=t2b5;;              #### times _q^(-1/2)
t21b6:= qcom2(t2b6,t2b7);; #### times _q^-1
t21b7:= _q * t2a1 * t2b1;; #### times _q^(-1/2)
t21a1:= t2a7;;
t21a2:= t2a1 *t2a2;;
t21a3:= t2a3;;
t21a5:= t2a5;;
t21a6:= t2a7* t2a6;;
t21a7:= t2a1;;
###################
t12b1:=qcom2(t1b1,t1b2);; #### times _q^-1
t12b2:= _q *t1a6 * t1b6;; #### times _q^(-1/2)
t12b3:=qcom2(t1b3,t1b2);; #### times _q^-1
t12b4:=t1b4;;
t12b5:=qcom2(t1b5,t1b6);; #### times _q^-1
t12b6:= _q * t1a2 *t1b2;; #### times _q^(-1/2)
t12b7:=qcom2(t1b7,t1b6);; #### times _q^-1
t12a1:= t1a1 * t1a2;;
t12a2:= t1a6;;
t12a3:= t1a2 * t1a3;;
t12a5:= t1a5 * t1a6;;
t12a6:= t1a2;;
t12a7:= t1a6 * t1a7;;
#####################
t121b1:= _q * t12a7 * t12b7;; #### times _q^-1
t121b2:= qcom2(t12b2,t12b1);; #### times _q^-2
t121b3:=t12b3;;              #### times _q^-1
t121b4:=t12b4;;
t121b5:=t12b5;;              #### times _q^-1
t121b6:= qcom2(t12b6,t12b7);; #### times _q^-2
t121b7:= _q * t12a1 * t12b1;; #### times _q^-1
t121a1:= t12a7;;
t121a2:= t12a1 *t12a2;;
t121a3:= t12a3;;
t121a5:= t12a5;;
t121a6:= t12a7* t12a6;;
t121a7:= t12a1;;
###################
t212b1:=qcom2(t21b1,t21b2);; #### times _q^-2
t212b2:= _q *t21a6 * t21b6;; #### times _q^-1
t212b3:=qcom2(t21b3,t21b2);; #### times _q^-2
t212b4:=t21b4;;
t212b5:=qcom2(t21b5,t21b6);; #### times _q^-2
t212b6:= _q * t21a2 *t21b2;; #### times _q^-1
t212b7:=qcom2(t21b7,t21b6);; #### times _q^-2
t212a1:= t21a1 * t21a2;;
t212a2:= t21a6;;
t212a3:= t21a2 * t21a3;;
t212a5:= t21a5 * t21a6;;
t212a6:= t21a2;;
t212a7:= t21a6 * t21a7;;
###############################################################################
########## Final step in the verification of the type A braid relation ########
########## between t1 and t2.                                        ########
########## We need to take into account the differences of q-factors   ########
###############################################################################
t121b1 - _q^-1 * t212b1;
t121b2 - _q * t212b2;
t121b3 - _q^-1 * t212b3;
t121b4 - t212b4;
t121b5 - _q^-1 * t212b5;
t121b6 - _q * t212b6;
t121b7 - _q^-1 * t212b7;
t121a1 - t212a1;
t121a2 - t212a2;
t121a3 - t212a3;
t121a5 - t212a5;
t121a6 - t212a6;
t121a7 - t212a7;
####### :-) End :-) #####################################