R:= RootSystem( "A",6);
 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];
 e1:=g[34]; e2:=g[36]; e3:=g[39]; e4:=g[43]; e5:=g[48]; e6:=g[54];
 k1:=g[22]; k2:=g[24]; k3:=g[26]; k4:=g[28]; k5:=g[30]; k6:=g[32];
 k1m:=g[23]; k2m:=g[25]; k3m:=g[27]; k4m:=g[29]; k5m:=g[31]; k6m:=g[33];   
########## generators of B ###############
 b1:=f1 - k1m * e6;
 b2:=f2 - k2m * e5;
 b3:=f3 - k3m * e4;
 b4:=f4 - k4m * e3;
 b5:=f5 - k5m * e2;
 b6:=f6 - k6m * e1;
 a1:= k1 * k6m;
 a2:= k2 * k5m;
 a3:= k3 * k4m;
 a4:= k4 * k3m;
 a5:= k5 * k2m;
 a6:= k6 * k1m;
########## q-numbers #################### 
 2q:=_q+_q^-1;
########## q-commutator #################
 qcom:=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 * b6 - b6 * b1) - a1 + a6;
(_q-_q^-1)*(b2 * b5 - b5 * b2) - a2 + a5;
rel1(b1,b2);
rel1(b2,b1);
b1 *b3 - b3* b1;
b2 * b4 - b4 * b2;
rel1(b2,b3);
rel1(b3,b2);
rel1(b3,b4) - 2q * b3 *( _q^2* a4 + _q^-1 * a3); #### from here onwards it's symmetric
rel1(b4,b3) - 2q * b4 *( _q^2* a3 + _q^-1 * a4);
rel1(b4,b5);
rel1(b5,b4);
b5 *b3 - b3* b5;
b6 * b4 - b4 * b6; 
rel1(b5,b6);
rel1(b6,b5);
################# \tau_3^- braid group action #####################
t3mb1:=b1;;
t3mb2:=_q^-2*qcom(b4,qcom(b3,b2)) + a4 * b2;; ##### times _q^(1/2)
t3mb3:= _q^-2 * a4 * b3;; ##### times _q^(1/2)
t3mb4:= _q^-2 * a3 * b4;; ##### times _q^(1/2)
t3mb5:=_q^-2*qcom(b3,qcom(b4,b5)) + a3 * b5;; ##### times _q^(1/2)
t3mb6:=b6;;
t3ma1:=a1;;
t3ma2:=a2;;
t3ma3:=a3;;
t3ma4:=a4;;
t3ma5:=a5;;
t3ma6:=a6;;
################# t3m is an algebra homomorphism ############
(_q-_q^-1)*(t3mb1 * t3mb6 - t3mb6 * t3mb1) - t3ma1 + t3ma6;
(_q-_q^-1)* _q *(t3mb2 * t3mb5 - t3mb5 * t3mb2) - t3ma2 + t3ma5;
rel1(t3mb1,t3mb2);
rel1(t3mb2,t3mb1);
t3mb1 *t3mb3 - t3mb3* t3mb1;
t3mb2 * t3mb4 - t3mb4 * t3mb2;
rel1(t3mb2,t3mb3);
rel1(t3mb3,t3mb2);
_q * rel1(t3mb3,t3mb4) - 2q * t3mb3 *( _q^2* t3ma4 + _q^-1 * t3ma3); #### from here onwards it's symmetric
################# the inverse \tau_3 of \tau_3^- ##################
t3b1:=b1;;
t3b2:=_q^-1*qcom(qcom(b2,b3),b4) + a3 * b2;; ##### times _q^(-1/2)
t3b3:=_q^2 * a3 * b3;;  ##### times _q^(-1/2)
t3b4:=_q^2 * a4 * b4;;  ##### times _q^(-1/2)
t3b5:=_q^-1*qcom(qcom(b5,b4),b3) + a4 * b5;; ##### times _q^(-1/2)
t3b6:=b6;;
t3a1:=a1;;
t3a2:=a2;;
t3a3:=a3;;
t3a4:=a4;;
t3a5:=a5;;
t3a6:=a6;;
################ t3 is an algebra homomorphism ################
(_q-_q^-1)*(t3b1 * t3b6 - t3b6 * t3b1) - t3a1 + t3a6;
(_q-_q^-1)* _q^-1 *(t3b2 * t3b5 - t3b5 * t3b2) - t3a2 + t3a5;
rel1(t3b1,t3b2);
rel1(t3b2,t3b1);
t3b1 *t3b3 - t3b3* t3b1;
t3b2 * t3b4 - t3b4 * t3b2;
rel1(t3b2,t3b3);
rel1(t3b3,t3b2);
_q^-1 * rel1(t3b3,t3b4) - 2q * t3b3 *( _q^2* t3a4 + _q^-1 * t3a3); #### from here onwards it's symmetric
########### t3 is the inverse of t3m ####################
qcom(qcom(t3mb2,t3mb3),t3mb4) + t3ma3 * t3mb2 - b2;
_q^2 * t3ma3 * t3mb3 - b3;
_q^2 * t3ma4 * t3mb4 - b4;
qcom(qcom(t3mb5,t3mb4),t3mb3) + t3ma4 * t3mb5 - b5;
_q^-3*qcom(t3b4,qcom(t3b3,t3b2)) + t3a4 * t3b2 - b2; 
_q^-2 * t3a4 * t3b3 - b3;
_q^-2 * t3a3 * t3b4 - b4;
_q^-3*qcom(t3b3,qcom(t3b4,t3b5)) + t3a3 * t3b5 - b5;
############ \tau_2^- braid group action ####################
t2mb1:=qcom(b2,b1);; #### times _q^(-1/2)
t2mb2:= _q^-1*a2 * b5;;
t2mb3:=qcom(b2,b3);; #### times _q^(-1/2)
t2mb4:=qcom(b5,b4);; #### times _q^(-1/2)
t2mb5:= _q^-1 * a5 *b2;;
t2mb6:=qcom(b5,b6);; #### times _q^(-1/2)
t2ma1:= a1 * a2;;
t2ma2:= a5;;
t2ma3:= a2 * a3;;
t2ma4:= a5 * a4;;
t2ma5:= a2;;
t2ma6:= a6 * a5;;
################# t2m is an algebra homomorphism ############
(_q-_q^-1)*_q^-1 * (t2mb1 * t2mb6 - t2mb6 * t2mb1) - t2ma1 + t2ma6;
(_q-_q^-1)*(t2mb2 * t2mb5 - t2mb5 * t2mb2) - t2ma2 + t2ma5;
rel1(t2mb1,t2mb2);
rel1(t2mb2,t2mb1);
t2mb1 *t2mb3 - t2mb3* t2mb1;
t2mb2 * t2mb4 - t2mb4 * t2mb2;
rel1(t2mb2,t2mb3);
rel1(t2mb3,t2mb2);
_q^-1 *rel1(t2mb3,t2mb4) - 2q * t2mb3 *( _q^2* t2ma4 + _q^-1 * t2ma3); #### from here onwards it's symmetric
########### the inverse \tau_2 of \tau_2^- #################
t2b1:= qcom(b1,b2);;  #### times _q^(-1/2)
t2b2:= _q *a5 * b5;;
t2b3:= qcom(b3,b2);;  #### times _q^(-1/2)
t2b4:= qcom(b4,b5);;  #### times _q^(-1/2)
t2b5:= _q * a2 *b2;;
t2b6:= qcom(b6,b5);;  #### times _q^(-1/2)
t2a1:= a1 * a2;;
t2a2:= a5;;
t2a3:= a2 * a3;;
t2a4:= a4 * a5;;
t2a5:= a2;;
t2a6:= a6 * a5;;
################# t2 is an algebra homomorphism ############
(_q-_q^-1)*_q^-1 * (t2b1 * t2b6 - t2b6 * t2b1) - t2a1 + t2a6;
(_q-_q^-1)*(t2b2 * t2b5 - t2b5 * t2b2) - t2a2 + t2a5;
rel1(t2b1,t2b2);
rel1(t2b2,t2b1);
t2b1 * t2b3 - t2b3 * t2b1;
t2b2 * t2b4 - t2b4 * t2b2;
rel1(t2b2,t2b3);
rel1(t2b3,t2b2);
_q^-1 * rel1(t2b3,t2b4) - 2q * t2b3 *( _q^2* t2a4 + _q^-1 * t2a3); #### from here onwards it's symmetric
########### t2m is the inverse of t2 #####################
_q^-1 * qcom(t2b2,t2b1) - b1;
_q^-1 * t2a2 * t2b5 - b2;
_q^-1 * qcom(t2b2,t2b3) - b3;
_q^-1 * qcom(t2b5,t2b4) - b4;
_q^-1 * t2a5 *t2b2 - b5;
_q^-1 * qcom(t2b5,t2b6) - b6; 
t2a1 * t2a2 - a1;
t2a5 - a2;
t2a2 * t2a3 - a3;
t2a5 * t2a4 - a4;
t2a2 - a5;
t2a6 * t2a5 - a6;
############ \tau_1^- braid group action ####################
t1mb1:= _q^-1 * a1 * b6;;
t1mb2:=qcom(b1,b2);; #### times _q^(-1/2);
t1mb3:=b3;;
t1mb4:=b4;;
t1mb5:=qcom(b6,b5);; #### times _q^(-1/2);
t1mb6:= _q^-1 * a6 * b1;;
t1ma1:= a6;;
t1ma2:= a1 * a2;;
t1ma3:= a3;;
t1ma4:= a4;;
t1ma5:= a6 * a5;;
t1ma6:= a1;;
################# t1m is an algebra homomorphism ############
(_q-_q^-1) * (t1mb1 * t1mb6 - t1mb6 * t1mb1) - t1ma1 + t1ma6;
(_q-_q^-1)* _q^-1 *(t1mb2 * t1mb5 - t1mb5 * t1mb2) - t1ma2 + t1ma5;
rel1(t1mb1,t1mb2);
rel1(t1mb2,t1mb1);
t1mb1 *t1mb3 - t1mb3* t1mb1;
t1mb2 * t1mb4 - t1mb4 * t1mb2;
rel1(t1mb2,t1mb3);
rel1(t1mb3,t1mb2);
rel1(t1mb3,t1mb4) - 2q * t1mb3 *( _q^2* t1ma4 + _q^-1 * t1ma3); #### from here onwards it's symmetric
########### the inverse \tau_1 of \tau_1^- #################
t1b1:= _q * a6 * b6;;
t1b2:=qcom(b2,b1);; #### times _q^(-1/2);
t1b3:=b3;;
t1b4:=b4;;
t1b5:=qcom(b5,b6);; #### times _q^(-1/2);
t1b6:= _q * a1 * b1;;
t1a1:= a6;;
t1a2:= a1 * a2;;
t1a3:= a3;;
t1a4:= a4;;
t1a5:= a6 * a5;;
t1a6:= a1;;
################# t1 is an algebra homomorphism ############
(_q-_q^-1) * (t1b1 * t1b6 - t1b6 * t1b1) - t1a1 + t1a6;
(_q-_q^-1) * _q^-1 *(t1b2 * t1b5 - t1b5 * t1b2) - t1a2 + t1a5;
rel1(t1b1,t1b2);
rel1(t1b2,t1b1);
t1b1 * t1b3 - t1b3 * t1b1;
t1b2 * t1b4 - t1b4 * t1b2;
rel1(t1b2,t1b3);
rel1(t1b3,t1b2);
rel1(t1b3,t1b4) - 2q * t1b3 *( _q^2* t1a4 + _q^-1 * t1a3); #### from here onwards it's symmetric
################ t1 is the inverse of t1m ##################
_q * t1ma6 * t1mb6 - b1;
_q^-1 * qcom(t1mb2,t1mb1)-b2; 
_q^-1 * qcom(t1mb5,t1mb6)-b5;
_q * t1ma1 * t1mb1 - b6;
_q^-1 * t1a1 * t1b6 - b1;
_q^-1 * qcom(t1b1,t1b2) - b2; 
_q^-1 * qcom(t1b6,t1b5) - b5;
_q^-1 * t1a6 * t1b1 - b6;
#############################################################
################ Verifying the braid relation ###############
################ t2 t1 t2 = t1 t2 t1          ###############
#############################################################
t21b1:= _q * t2a6 * t2b6;;  #### times _q^(-1/2);
t21b2:=qcom(t2b2,t2b1);;    #### times _q^(-1);
t21b3:=t2b3;;               #### times _q^(-1/2);
t21b4:=t2b4;;               #### times _q^(-1/2);
t21b5:=qcom(t2b5,t2b6);;    #### times _q^(-1);
t21b6:= _q * t2a1 * t2b1;;  #### times _q^(-1/2);
t21a1:= t2a6;;
t21a2:= t2a1 * t2a2;;
t21a3:= t2a3;;
t21a4:= t2a4;;
t21a5:= t2a6 * t2a5;;
t21a6:= t2a1;;
#####
t12b1:= qcom(t1b1,t1b2);;  #### times _q^(-1)
t12b2:= _q *t1a5 * t1b5;;  #### times _q^(-1/2)
t12b3:= qcom(t1b3,t1b2);;  #### times _q^(-1)
t12b4:= qcom(t1b4,t1b5);;  #### times _q^(-1)
t12b5:= _q * t1a2 *t1b2;;  #### times _q^(-1/2)
t12b6:= qcom(t1b6,t1b5);;  #### times _q^(-1)
t12a1:= t1a1 * t1a2;;
t12a2:= t1a5;;
t12a3:= t1a2 * t1a3;;
t12a4:= t1a4 * t1a5;;
t12a5:= t1a2;;
t12a6:= t1a6 * t1a5;;
#####
t121b1:= _q * t12a6 * t12b6;;  #### times _q^(-1);
t121b2:=qcom(t12b2,t12b1);;    #### times _q^(-2);
t121b3:=t12b3;;               #### times _q^(-1);
t121b4:=t12b4;;               #### times _q^(-1);
t121b5:=qcom(t12b5,t12b6);;    #### times _q^(-2);
t121b6:= _q * t12a1 * t12b1;;  #### times _q^(-1);
#####
t212b1:= qcom(t21b1,t21b2);;  #### times _q^(-2)
t212b2:= _q *t21a5 * t21b5;;  #### times _q^(-1)
t212b3:= qcom(t21b3,t21b2);;  #### times _q^(-2)
t212b4:= qcom(t21b4,t21b5);;  #### times _q^(-2)
t212b5:= _q * t21a2 *t21b2;;  #### times _q^(-1)
t212b6:= qcom(t21b6,t21b5);;  #### times _q^(-2)
############# final check ###################################
t121b1 - _q^-1 * t212b1;
_q^-1 * t121b2 - t212b2;
t121b3 - _q^-1 * t212b3;
t121b4 - _q^-1 * t212b4;
_q^-1 * t121b5 - t212b5;
t121b6 - _q^-1 * t212b6;
#############################################################
################ Verifying the braid relation ###############
################ t2 t3 t2 t3 = t3 t2 t3 t2    ###############
#############################################################
t3b1:=b1;;
t3b2:=_q^-1*qcom(qcom(b2,b3),b4) + a3 * b2;; ##### times _q^(-1/2)
t3b3:=_q^2 * a3 * b3;;  ##### times _q^(-1/2)
t3b4:=_q^2 * a4 * b4;;  ##### times _q^(-1/2)
t3b5:=_q^-1*qcom(qcom(b5,b4),b3) + a4 * b5;; ##### times _q^(-1/2)
t3b6:=b6;;
t3a1:=a1;;
t3a2:=a2;;
t3a3:=a3;;
t3a4:=a4;;
t3a5:=a5;;
t3a6:=a6;;
##############
t2b1:= qcom(b1,b2);;  #### times _q^(-1/2)
t2b2:= _q *a5 * b5;;
t2b3:= qcom(b3,b2);;  #### times _q^(-1/2)
t2b4:= qcom(b4,b5);;  #### times _q^(-1/2)
t2b5:= _q * a2 *b2;;
t2b6:= qcom(b6,b5);;  #### times _q^(-1/2)
t2a1:= a1 * a2;;
t2a2:= a5;;
t2a3:= a2 * a3;;
t2a4:= a4 * a5;;
t2a5:= a2;;
t2a6:= a6 * a5;;
##############
t23b1:=t2b1;; #### times _q^(-1/2)
t23b2:=_q^-2*qcom(qcom(t2b2,t2b3),t2b4) + t2a3 * t2b2;; ##### times _q^(-1/2)
t23b3:=_q * t2a3 * t2b3;;  
t23b4:=_q * t2a4 * t2b4;;  
t23b5:=_q^-2*qcom(qcom(t2b5,t2b4),t2b3) + t2a4 * t2b5;; ##### times _q^(-1/2)
t23b6:=t2b6;; #### times _q^(-1/2)
t23a1:=t2a1;;
t23a2:=t2a2;;
t23a3:=t2a3;;
t23a4:=t2a4;;
t23a5:=t2a5;;
t23a6:=t2a6;;
##############
t32b1:= _q^-1 * qcom(t3b1,t3b2);;
t32b2:= _q *t3a5 * t3b5;;  #### times _q^(-1/2)
t32b3:= _q^-1 * qcom(t3b3,t3b2);;  #### times _q^(-1/2)
t32b4:= _q^-1 * qcom(t3b4,t3b5);;  #### times _q^(-1/2)
t32b5:= _q * t3a2 *t3b2;;  #### times _q^(-1/2)
t32b6:= _q^-1 * qcom(t3b6,t3b5);;  
t32a1:= t3a1 * t3a2;;
t32a2:= t3a5;;
t32a3:= t3a2 * t3a3;;
t32a4:= t3a4 * t3a5;;
t32a5:= t3a2;;
t32a6:= t3a6 * t3a5;;
##############
t323b1:=t32b1;; #### times _q^(-1/2)
t323b2:=_q^-3*qcom(qcom(t32b2,t32b3),t32b4) + _q^-1 * t32a3 * t32b2;;
t323b3:=_q * t32a3 * t32b3;;  
t323b4:=_q * t32a4 * t32b4;;  
t323b5:=_q^-3*qcom(qcom(t32b5,t32b4),t32b3) + _q^-1 * t32a4 * t32b5;;
t323b6:=t32b6;; #### times _q^(-1/2)
t323a1:=t32a1;;
t323a2:=t32a2;;
t323a3:=t32a3;;
t323a4:=t32a4;;
t323a5:=t32a5;;
t323a6:=t32a6;;
##############
t232b1:= _q^-1 * qcom(t23b1,t23b2);; #### times _q^(-1/2)
t232b2:= _q *t23a5 * t23b5;;  #### times _q^(-1/2)
t232b3:= _q^-1 * qcom(t23b3,t23b2);;  
t232b4:= _q^-1 * qcom(t23b4,t23b5);;  
t232b5:= _q * t23a2 *t23b2;;  #### times _q^(-1/2)
t232b6:= _q^-1 * qcom(t23b6,t23b5);; #### times _q^(-1/2) 
t232a1:= t23a1 * t23a2;;
t232a2:= t23a5;;
t232a3:= t23a2 * t23a3;;
t232a4:= t23a4 * t23a5;;
t232a5:= t23a2;;
t232a6:= t23a6 * t23a5;;
##############
t2323b1:=t232b1;; #### times _q^-1
t2323b2:=_q^-2*qcom(qcom(t232b2,t232b3),t232b4) + _q^-1 * t232a3 * t232b2;; 
t2323b3:=_q * t232a3 * t232b3;;  #### times _q^(1/2)
t2323b4:=_q * t232a4 * t232b4;;  #### times _q^(1/2)
t2323b5:=_q^-2*qcom(qcom(t232b5,t232b4),t232b3) + _q^-1 * t232a4 * t232b5;;
t2323b6:=t232b6;; #### times _q^-1
t2323a1:=t232a1;;
t2323a2:=t232a2;;
t2323a3:=t232a3;;
t2323a4:=t232a4;;
t2323a5:=t232a5;;
t2323a6:=t232a6;;
##############
t3232b1:= _q^-1 * qcom(t323b1,t323b2);; 
t3232b2:= _q *t323a5 * t323b5;;  
t3232b3:= _q^-1 * qcom(t323b3,t323b2);;  #### times _q^(1/2)
t3232b4:= _q^-1 * qcom(t323b4,t323b5);;    #### times _q^(1/2)
t3232b5:= _q * t323a2 *t323b2;;  
t3232b6:= _q^-1 * qcom(t323b6,t323b5);;  
t3232a1:= t323a1 * t323a2;;
t3232a2:= t323a5;;
t3232a3:= t323a2 * t323a3;;
t3232a4:= t323a4 * t323a5;;
t3232a5:= t323a2;;
t3232a6:= t323a6 * t323a5;;
############ final check ##################
t3232b1 - _q^-1 * t2323b1;
t3232b2 -  t2323b2;
t3232b3 -  t2323b3;
t3232b4 -  t2323b4;
t3232b5 -  t2323b5;
t3232b6 - _q^-1 * t2323b6;
#######################


####### :-) End :-) #####################################