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];   
######## Inverse of Lusztig automorphisms ##################
tau:=AntiAutomorphismTau( U );
T1:=AutomorphismTalpha(U,1);
T2:=AutomorphismTalpha(U,2);
T3:=AutomorphismTalpha(U,3);
T4:=AutomorphismTalpha(U,4);
T5:=AutomorphismTalpha(U,5);
T6:=AutomorphismTalpha(U,6);
T7:=AutomorphismTalpha(U,7);
T1m:= tau*T1*tau;
T2m:= tau*T2*tau;
T3m:= tau*T3*tau;
T4m:= tau*T4*tau;
T5m:= tau*T5*tau;
T6m:= tau*T6*tau;
T7m:= tau*T7*tau;
########## generators of B ###############
 b1:=f1;
 b2:=f2-k2m*Image(T1*T3,e2);
 b3:=f3;
 b4:=f4-k4m*Image(T5*T3,e4);
 b5:=f5;
 b6:=f6-k6m*Image(T7*T5,e6);
 b7:=f7;
########## q-numbers #################### 
 2q:=_q+_q^-1;
########## q-commutator #################
 qcom2:=function(a,b)
   return a*b-_q*b*a;
 end;
 qcom3:=function(a,b,c)
   return qcom2(a,b)*c - _q * c *qcom2(a,b);
 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;
################ non-homogeneous relation  ########################
relEven1:=function(a,b,eb,ebpm,kbm,kb)
   return a^2*b - (_q + _q^-1)*a*b*a + b*a^2+_q^-1*(_q-_q^-1)^2*b*eb*ebpm+_q^-2*kbm*ebpm+ kb*ebpm;
 end;
################ lower order term of relEven1 #####################
lotEven1:=function(b,eb,ebpm,kbm,kb)
   return _q^-1*(_q-_q^-1)^2*b*eb*ebpm+_q^-2*kbm*ebpm+ kb*ebpm;
 end;
############################################################
########## confirming the relations of Proposition 5.2 #####
########## It suffices to check the relations not      #####
########## involving b6, b7, e6, e7, etc.              #####
############################################################
######### Vanishing commutators (using symmetry) ###########
############################################################
b2 * b4 - b4 * b2;
######### other q-Serre type relations ##########################
rel1(b1,b2);
rel1(b3,b2);
rel1(b2,b1)+ lotEven1(b1,e1,e3,k1m,k1);
rel1(b2,b3)+ lotEven1(b3,e3,e1,k3m,k3);
#############################################################
################ T4m T3m T5m T4m (-k4m T3 T5(e4)) ###########
################ verifying (5.4)                  ###########
#############################################################
Image(T4m*T3m*T5m*T4m, -k4m*Image(T3*T5,e4)) -
((_q-_q^-1)^2 * qcom3(f4,f5,f3) * e3*e5 
 - _q^2*(_q-_q^-1) * qcom2(f4,f5)*k3*e5 
 - _q^2*(_q-_q^-1) * qcom2(f4,f3)*k5*e3 + _q^4 *f4*k3*k5);
#############################################################
################# tau1^- Braid group action #####################
#############################################################
t1mb1:= b3;;
t1mb2:= _q^3 * b2 *k1 * k3 + _q^-1*(_q-_q^-1)^2 *qcom3(b2,b1,b3)*e1*e3 - _q*(_q-_q^-1)*(qcom2(b2,b3)*k1*e3 + qcom2(b2,b1)*k3*e1);;
t1mb3:= b1;;
t1mb4:= qcom3(b2,b3,b4);;   #### factor q^{-1/2} missing !! ####
t1mb5:= b5;;
t1mb6:= b6;;
t1mb7:= b7;;
################# tau2^- Braid group action #########################
t2mb1:= b1;;
t2mb2:= qcom3(b4,b3,b2);;   #### factor q^{-1/2} missing !! ####
t2mb3:= b5;;
t2mb4:= _q^3 * b4 *k3 * k5 + _q^-1*(_q-_q^-1)^2 *qcom3(b4,b3,b5)*e3*e5 - _q*(_q-_q^-1)*(qcom2(b4,b5)*k3*e5 + qcom2(b4,b3)*k5*e3) ;;
t2mb5:= b3;;
t2mb6:= qcom3(b4,b5,b6);;   #### factor q^{-1/2} missing !! ####
t2mb7:= b7;;
################# tau3^- Braid group action #########################
t3mb1:= b1;;
t3mb2:= b2;;
t3mb3:= b3;;
t3mb4:= qcom3(b6,b5,b4);;   #### factor q^{-1/2} missing !! ####
t3mb5:= b7;;
t3mb6:= _q^3 * b6 *k5 * k7 + _q^-1*(_q-_q^-1)^2 *qcom3(b6,b5,b7)*e5*e7 - _q*(_q-_q^-1)*(qcom2(b6,b7)*k5*e7 + qcom2(b6,b5)*k7*e5) ;;
t3mb7:= b5;;
###################################################################################
#### The elment t2mb4 is obtained by evaluating Image(T4m*T3m*T5m*T6m,b4) and   ###
#### replacing F by B in the leading terms -- moreover we rescale with factor q ###
###################################################################################
###############################################################
## Observe that t2mb2 = b4*Image(T3m,b2)-_q*Image(T3m,b2)*b4.
## Also, t1mb4 coincides with Image(T2m*T1m*T3m*T2m,b4) up to terms with nontrivial
## U^+ component. (And we need to rescale!)
###############################################################

#####################################################################
#### We now show that t1m, t2m, t3m are algebra homomorphisms.  #####
#### It suffices to show that t2m is an algebra homomorphism.   #####
#### It is clear that the restriction of t2m to the subalgebra  #####
#### generated by all ei, fi, ki, and kim for odd i is an       #####
#### homomorphism. It hence suffices to check that the relations#####
#### given in Proposition 5.2 are preserved under t2m.          #####
#### This is done in the following.                             ##### 
#####################################################################
######### ki commutators for odd i ##################################
k1 * t2mb2 - _q * t2mb2 * k1;
k3 * t2mb2 -      t2mb2 * k3;
k5 * t2mb2 - _q * t2mb2 * k5;
k7 * t2mb2 -      t2mb2 * k7;
k1 * t2mb4 -      t2mb4 * k1;
k3 * t2mb4 - _q * t2mb4 * k3;
k5 * t2mb4 - _q * t2mb4 * k5;
k7 * t2mb4 -      t2mb4 * k7;
k1 * t2mb6 -      t2mb6 * k1;
k3 * t2mb6 - _q * t2mb6 * k3;
k5 * t2mb6 -      t2mb6 * k5;
k7 * t2mb6 - _q * t2mb6 * k7;
######### ei commutators for odd i ################################
e1 * t2mb2 -  t2mb2 * e1;
e3 * t2mb2 -  t2mb2 * e3;
e5 * t2mb2 -  t2mb2 * e5;
e7 * t2mb2 -  t2mb2 * e7;
e1 * t2mb4 -  t2mb4 * e1;
e3 * t2mb4 -  t2mb4 * e3;
e5 * t2mb4 -  t2mb4 * e5;
e7 * t2mb4 -  t2mb4 * e7;
e1 * t2mb6 -  t2mb6 * e1; 
e3 * t2mb6 -  t2mb6 * e3;
e5 * t2mb6 -  t2mb6 * e5;
e7 * t2mb6 -  t2mb6 * e7;
######### Vanishing commutators ############################
t2mb2 * t2mb4 - t2mb4 * t2mb2;
t2mb2 * t2mb6 - t2mb6 * t2mb2;
t2mb4 * t2mb6 - t2mb6 * t2mb4;
######### other q-Serre type relations ##########################
rel1(t2mb1,t2mb2);
rel1(t2mb3,t2mb2);
t2mb5*t2mb2 - t2mb2* t2mb5;
t2mb7*t2mb2 - t2mb2* t2mb7;
rel1(t2mb2,t2mb1)+ _q* lotEven1(t2mb1,e1,e5,k1m,k1);
rel1(t2mb2,t2mb3)+ _q* lotEven1(t2mb3,e5,e1,k5m,k5);
rel1(t2mb3,t2mb4);
rel1(t2mb5,t2mb4);
rel1(t2mb4,t2mb3) + lotEven1(t2mb3,e5,e3,k5m,k5);
rel1(t2mb4,t2mb5) + lotEven1(t2mb5,e3,e5,k3m,k3);
t2mb1*t2mb4 - t2mb4* t2mb1;
t2mb7*t2mb4 - t2mb4* t2mb7;
#################################################################

###################################mmm#######################################
#### To show that t1m, t2m, and t3m are algebra automorphisms we need to ####
#### give their inverses. The inverses were obtained by trial and error. ####
#### Again, it suffices to check everything for t2, the inverse of t2m.  #### 
#############################################################################
######## t1, the inverse of t1m #########################
t1b1:=b3;;
t1b2:= _q^-3*b2*k1m*k3m +_q^-1*(_q-_q^-1)^2*qcom2(b3,qcom2(b1,b2))*e1*e3 
        - _q^-2*(_q-_q^-1)*(qcom2(b3,b2)*k1m*e3 + qcom2(b1,b2)*k3m*e1);;
t1b3:= b1;;
t1b4:= qcom3(b4,b3,b2);;   #### factor q^{-1/2} missing !! ####
t1b5:= b5;;
t1b6:= b6;;
t1b7:= b7;;
######## t2, the inverse of t2m ##########################
t2b1:=b1;;
t2b2:= qcom3(b2,b3,b4);;   #### factor q^{-1/2} missing !! ####
t2b3:=b5;;
t2b4:= _q^-3*b4*k3m*k5m +_q^-1*(_q-_q^-1)^2*qcom2(b5,qcom2(b3,b4))*e3*e5 
        - _q^-2*(_q-_q^-1)*(qcom2(b5,b4)*k3m*e5 + qcom2(b3,b4)*k5m*e3);;
t2b5:= b3;;
t2b6:= qcom3(b6,b5,b4);;   #### factor q^{-1/2} missing !! ####
t2b7:= b7;;
######## t3, the inverse of t3m ##########################
t3b1:=b1;;
t3b2:=b2;;
t3b3:=b3;;
t3b4:= qcom3(b4,b5,b6);;   #### factor q^{-1/2} missing !! ####
t3b5:=b7;;
t3b6:= _q^-3*b6*k5m*k7m +_q^-1*(_q-_q^-1)^2*qcom2(b7,qcom2(b5,b6))*e5*e7 
        - _q^-2*(_q-_q^-1)*(qcom2(b7,b6)*k5m*e7 + qcom2(b5,b6)*k7m*e5);;
t3b7:= b5;;
######### t2 is an algebra homomorphism ############################
######### ki commutators for odd i #################################
k1 * t2b2 - _q * t2b2 * k1;
k3 * t2b2 -      t2b2 * k3;
k5 * t2b2 - _q * t2b2 * k5;
k7 * t2b2 -      t2b2 * k7;
k1 * t2b4 -      t2b4 * k1;
k3 * t2b4 - _q * t2b4 * k3;
k5 * t2b4 - _q * t2b4 * k5;
k7 * t2b4 -      t2b4 * k7;
k1 * t2b6 -      t2b6 * k1;
k3 * t2b6 - _q * t2b6 * k3;
k5 * t2b6 -      t2b6 * k5;
k7 * t2b6 - _q * t2b6 * k7;
######### ei commutators for odd i #################################
e1 * t2b2 -  t2b2 * e1;
e3 * t2b2 -  t2b2 * e3;
e5 * t2b2 -  t2b2 * e5;
e7 * t2b2 -  t2b2 * e7;
e1 * t2b4 -  t2b4 * e1;
e3 * t2b4 -  t2b4 * e3;
e5 * t2b4 -  t2b4 * e5;
e7 * t2b4 -  t2b4 * e7;
e1 * t2b6 -  t2b6 * e1; 
e3 * t2b6 -  t2b6 * e3;
e5 * t2b6 -  t2b6 * e5;
e7 * t2b6 -  t2b6 * e7;
######### Vanishing commutators ############################
t2b2 * t2b4 - t2b4 * t2b2;
t2b2 * t2b6 - t2b6 * t2b2;
t2b4 * t2b6 - t2b6 * t2b4;
######### other q-Serre type relations ##########################
rel1(t2b1,t2b2);
rel1(t2b3,t2b2);
t2b5*t2b2 - t2b2* t2b5;
t2b7*t2b2 - t2b2* t2b7;
rel1(t2b2,t2b1)+ _q* lotEven1(t2b1,e1,e5,k1m,k1);
rel1(t2b2,t2b3)+ _q* lotEven1(t2b3,e5,e1,k5m,k5);
rel1(t2b3,t2b4);
rel1(t2b5,t2b4);
rel1(t2b4,t2b3) + lotEven1(t2b3,e5,e3,k5m,k5);
rel1(t2b4,t2b5) + lotEven1(t2b5,e3,e5,k3m,k3);
t2b1*t2b4 - t2b4* t2b1;
t2b7*t2b4 - t2b4* t2b7;
################################################################
#### We now check that t2 is indeed the inverse of t2m. ########
################################################################
###### t2 t2m = id ########
t2b1-b1;
_q^-1*qcom3(t2b4,t2b3,t2b2)-b2; 
t2b5-b3;
_q^3 * t2b4 *k3 * k5 + _q^-1*(_q-_q^-1)^2 *qcom3(t2b4,t2b3,t2b5)*e3*e5 
    -_q*(_q-_q^-1)*(qcom2(t2b4,t2b5)*k5*e3+qcom2(t2b4,t2b3)*k3*e5)-b4 ;
t2b3 - b5;
_q^-1* qcom3(t2b4,t2b5,t2b6) -b6; 
t2b7-b7; 
###### t2m t2 = id #########
t2mb1-b1;
_q^-1*qcom3(t2mb2,t2mb3,t2mb4) - b2;
t2mb5 - b3;
_q^-3*t2mb4*k3m*k5m+_q^-1*(_q-_q^-1)^2*qcom2(t2mb5,qcom2(t2mb3,t2mb4))*e3*e5 
  -_q^-2*(_q-_q^-1)*(qcom2(t2mb5,t2mb4)*k5m*e3+qcom2(t2mb3,t2mb4)*k3m*e5)-b4;
t2mb3-b5;
_q^-1*qcom3(t2mb6,t2mb5,t2mb4)-b6;
t2mb7-b7;
#################################################################
##################### Towards braid relations ###################
##############################################################################
###### In the following we verify the braid relation t2 t1 t2 = t1 t2 t1. ####
###### The relation t2 t3 t2 = t3 t2 t3 then holds for symmetry reasons.  ####
###### Furthermore, we verify that t1 t3 = t3 t1. As we already know that ####
###### the ti are algebra homomorphisms it suffices to check these        ####
###### relations on the generators bi and on the elements ej, fj, and kj. ####
###### The braid relation t2 t1 t2(ej) = t1 t2 t1(ej), for odd j, is      ####
###### checked by hand, and similarly for fj, kj, and kjm. Hence we only  ####
###### need to check t2 t1 t2(bi) = t1 t2 t1(bi) for all even i.          ####
###### Similarly, we only need to check that t1 t3(bi)=t3 t1(bi) for      ####
###### even i.                                                            ####
###### We use the notation t121b4 for t1 t2 t1(b4), and analogous         ####
###### notation for other braid group actions on the genertors bi.        ####  
############################################################################## 
######### Braid relation preparations #################################
t23b1:=t2b1;;
t23b2:=t2b2;;
t23b3:=t2b3;;
t23b4:= qcom3(t2b4,t2b5,t2b6);;   #### factor q^{-1} missing !! ####
t23b5:=t2b7;;
t23b6:=_q^-3*t2b6*k3m*k7m +_q^-1*(_q-_q^-1)^2*qcom2(t2b7,qcom2(t2b5,t2b6))*e3*e7 
        - _q^-2*(_q-_q^-1)*(qcom2(t2b7,t2b6)*k3m*e7 + qcom2(t2b5,t2b6)*k7m*e3);;
t23b7:= t2b5;;
############
t32b1:=t3b1;;
t32b2:= qcom3(t3b2,t3b3,t3b4);;   #### factor q^{-1} missing !! ####
t32b3:=t3b5;;
t32b4:=_q^-3*t3b4*k3m*k7m +_q^-1*(_q-_q^-1)^2*qcom2(t3b5,qcom2(t3b3,t3b4))*e3*e7 
        - _q^-2*(_q-_q^-1)*(qcom2(t3b5,t3b4)*k3m*e7 + qcom2(t3b3,t3b4)*k7m*e3);;
t32b5:= t3b3;;
t32b6:= qcom3(t3b6,t3b5,t3b4);;   #### factor q^{-1} missing !! ####
t32b7:= t3b7;;
############
t323b1:=t32b1;;
t323b2:=t32b2;;
t323b3:=t32b3;;
t323b4:= qcom3(t32b4,t32b5,t32b6);;   #### factor q^{-1} missing !! ####
t323b5:=t32b7;;
t323b6:=_q^-3*t32b6*k3m*k5m +_q^-1*(_q-_q^-1)^2*qcom2(t32b7,qcom2(t32b5,t32b6))*e3*e5 
        - _q^-2*(_q-_q^-1)*(qcom2(t32b7,t32b6)*k3m*e5 + qcom2(t32b5,t32b6)*k5m*e3);;
t323b7:= t32b5;;
############
t232b1:=t23b1;;
t232b2:= qcom3(t23b2,t23b3,t23b4);;   #### factor q^{-1} missing !! ####
t232b3:=t23b5;;
t232b4:=_q^-3*t23b4*k5m*k7m +_q^-1*(_q-_q^-1)^2*qcom2(t23b5,qcom2(t23b3,t23b4))*e5*e7 
        - _q^-2*(_q-_q^-1)*(qcom2(t23b5,t23b4)*k5m*e7 + qcom2(t23b3,t23b4)*k7m*e5);;
t232b5:= t23b3;;
t232b6:= qcom3(t23b6,t23b5,t23b4);;   #### factor q^{-1} missing !! ####
t232b7:= t23b7;;
############ Now we check the braid relation t232 = t323 ##############
t232b1 - t323b1;
_q^-1 * t232b2 - t323b2;
t232b3 - t323b3;
t232b4 - _q^-1 * t323b4;
t232b5 - t323b5;
_q^-1 * t232b6 - t323b6;
t232b7 - t323b7;
#######################################################################
######## More braid relations preparations ############################
t31b1:= t3b3;;
t31b2:= _q^-3*t3b2*k1m*k3m +_q^-1*(_q-_q^-1)^2*qcom2(t3b3,qcom2(t3b1,t3b2))*e1*e3 
        - _q^-2*(_q-_q^-1)*(qcom2(t3b3,t3b2)*k1m*e3 + qcom2(t3b1,t3b2)*k3m*e1);;
t31b3:= t3b1;;
t31b4:= qcom3(t3b4,t3b3,t3b2);;   #### factor q^{-1} missing !! ####
t31b5:= t3b5;;
t31b6:= t3b6;;
t31b7:= t3b7;;
########
t13b1:=t1b1;;
t13b2:=t1b2;;
t13b3:=t1b3;;
t13b4:= qcom3(t1b4,t1b5,t1b6);;   #### factor q^{-1} missing !! ####
t13b5:=t1b7;;
t13b6:= _q^-3*t1b6*k5m*k7m +_q^-1*(_q-_q^-1)^2*qcom2(t1b7,qcom2(t1b5,t1b6))*e5*e7 
        - _q^-2*(_q-_q^-1)*(qcom2(t1b7,t1b6)*k5m*e7 + qcom2(t1b5,t1b6)*k7m*e5);;
t13b7:= t1b5;;
######### Now we check the braid relations t13 = t31 ##################
t13b1 - t31b1;
t13b2 - t31b2;
t13b3 - t31b3;
t13b4 - t31b4;
t13b5 - t31b5;
t13b6 - t31b6;
t13b7 - t31b7;
######################################################################

################## Semidirect product relations ###########################
#### We are checking here the relations in Section 5.2 for i=2 ########
######################################################################
###### Relation (5.16) #####
qcom2(t2b3,t2b4) - Image(T5m,t2b4);
###### Relation (5.17) #####
t2b6 - Image(T5m,t2b6);
########################################################

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