R:= RootSystem( "D",5);
 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[20];
 e1:=g[31]; e2:=g[33]; e3:=g[36]; e4:=g[40]; e5:=g[50];
 k1:=g[21]; k2:=g[23]; k3:=g[25]; k4:=g[27]; k5:=g[29]; 
 k1m:=g[22]; k2m:=g[24]; k3m:=g[26]; k4m:=g[28]; k5m:=g[30];   
########## generators of B ###############
 b1:=f1 - k1m * e1;
 b2:=f2 - k2m * e2;
 b3:=f3 - k3m * e3;
 b4:=f4 - k4m * e5;
 b5:=f5 - k5m * e4;
 a4:= k4 * k5m;
 a5:= k5 * k4m;
########## 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)*(b4 * b5 - b5 * b4) - a4 + a5;
rel1(b2,b3) + _q^-1*b3;
rel1(b3,b2) + _q^-1*b2;
rel1(b4,b3);
rel1(b3,b4) + _q^-1 * b4;
b2*b4-b4*b2;
b2*b5-b5*b2;
rel1(b5,b3);
rel1(b3,b5) + _q^-1 * b5; 
################# \tau_2^{-1} Braid group action #####################
t2mb1:=qcom2(b2,b1);;
t2mb2:=b2;;
t2mb3:=qcom2(b2,b3);; 
t2mb4:=b4;;
t2mb5:=b5;;
t2ma4:=a4;;
t2ma5:=a5;;
################# t2m is an algebra homomorphism ############
(_q-_q^-1)* (t2mb4 * t2mb5 - t2mb5 * t2mb4) - t2ma4 + t2ma5;
rel1(t2mb2,t2mb3) + _q^-1 * t2mb3;
rel1(t2mb3,t2mb2) + _q^-1 * t2mb2;
rel1(t2mb4,t2mb3);
rel1(t2mb3,t2mb4) + _q^-1 * t2mb4;
t2mb2*t2mb4 - t2mb4*t2mb2;
t2mb2*t2mb5 - t2mb5*t2mb2;
rel1(t2mb5,t2mb3);
rel1(t2mb3,t2mb5) + _q^-1 * t2mb5;  
################# the inverse \tau_2 of \tau_2^{-1} #####################
t2b1:=qcom2(b1,b2);;
t2b2:=b2;;
t2b3:=qcom2(b3,b2);; 
t2b4:=b4;;
t2b5:=b5;;
t2a4:=a4;;
t2a5:=a5;;
################# t2 is an algebra homomorphism ############
(_q-_q^-1)* (t2b4 * t2b5 - t2b5 * t2b4) - t2a4 + t2a5;
rel1(t2b2,t2b3) + _q^-1 * t2b3;
rel1(t2b3,t2b2) + _q^-1 * t2b2;
rel1(t2b4,t2b3);
rel1(t2b3,t2b4) + _q^-1 * t2b4;
t2b2*t2b4 - t2b4*t2b2;
t2b2*t2b5 - t2b5*t2b2;
rel1(t2b5,t2b3);
rel1(t2b3,t2b5) + _q^-1 * t2b5; 
################ t2m is the inverse of t2 ######################
qcom2(t2b2,t2b1) - b1;
qcom2(t2b2,t2b3) - b3;
qcom2(t2mb1,t2mb2) - b1;
qcom2(t2mb3,t2mb2) - b3; 
################# \tau_3^{-1} Braid group action #####################
t3mb1:=b1;;
t3mb2:=qcom2(b3,b2);;
t3mb3:=b3;;
t3mb4:=qcom2(b3,b4);; 
t3mb5:=qcom2(b3,b5);;
t3ma4:=a4;;
t3ma5:=a5;;
################# t3m is an algebra homomorphism ############
(_q-_q^-1)* (t3mb4 * t3mb5 - t3mb5 * t3mb4) - t3ma4 + t3ma5;
rel1(t3mb2,t3mb3) + _q^-1 * t3mb3;
rel1(t3mb3,t3mb2) + _q^-1 * t3mb2;
rel1(t3mb4,t3mb3);
rel1(t3mb3,t3mb4) + _q^-1 * t3mb4;
t3mb2*t3mb4 - t3mb4*t3mb2;
t3mb2*t3mb5 - t3mb5*t3mb2;
rel1(t3mb5,t3mb3);
rel1(t3mb3,t3mb5) + _q^-1 * t3mb5;  
################# the inverse \tau_3 of \tau_3^{-1} ##################
t3b1:=b1;;
t3b2:=qcom2(b2,b3);;
t3b3:=b3;;
t3b4:=qcom2(b4,b3);; 
t3b5:=qcom2(b5,b3);; 
t3a4:=a4;;
t3a5:=a5;;
################# t3 is an algebra homomorphism ############
(_q-_q^-1)* (t3b4 * t3b5 - t3b5 * t3b4) - t3a4 + t3a5;
rel1(t3b2,t3b3) + _q^-1 * t3b3;
rel1(t3b3,t3b2) + _q^-1 * t3b2;
rel1(t3b4,t3b3);
rel1(t3b3,t3b4) + _q^-1 * t3b4;
t3b2*t3b4 - t3b4*t3b2;
t3b2*t3b5 - t3b5*t3b2;
rel1(t3b5,t3b3);
rel1(t3b3,t3b5) + _q^-1 * t3b5;  
############# t3m is the inverse of t3 ########################
qcom2(t3mb2,t3mb3) -b2;
qcom2(t3mb4,t3mb3) -b4; 
qcom2(t3mb5,t3mb3) -b5; 
qcom2(t3b3,t3b2) - b2;
qcom2(t3b3,t3b4) - b4; 
qcom2(t3b3,t3b5) - b5; 
############## \tau_4^{-1} Braid group action #####################
t4mb1:=b1;;
t4mb2:=b2;;
t4mb3:= _q^-1 * qcom2(b4,qcom2(b5,b3))+b3*a4;;
t4mb4:= _q^-1 * a4* b5;;
t4mb5:= _q^-1 * a5 * b4;;
t4ma4:= a5;;
t4ma5:= a4;;
############ t4m is an algebra homomorphism #############
(_q-_q^-1)* (t4mb4 * t4mb5 - t4mb5 * t4mb4) - t4ma4 + t4ma5;
rel1(t4mb2,t4mb3) + _q^-1 * t4mb3;
rel1(t4mb3,t4mb2) + _q^-1 * t4mb2;
rel1(t4mb4,t4mb3);
rel1(t4mb3,t4mb4) + _q^-1 * t4mb4;
t4mb2*t4mb4 - t4mb4*t4mb2;
t4mb2*t4mb5 - t4mb5*t4mb2;
rel1(t4mb5,t4mb3);
rel1(t4mb3,t4mb5) + _q^-1 * t4mb5;  
########### the inverse \tau_4 of \tau_4^{-1} #################
t4b1:= b1;;
t4b2:=b2;;
t4b3:= _q^-1 * qcom2(qcom2(b3,b4),b5)+ b3*a4;;
t4b4:= _q * a5 * b5;;
t4b5:= _q * a4 * b4;;
t4a4:= a5;;
t4a5:= a4;;
############ t4 is an algebra homomorphism #############
(_q-_q^-1)* (t4b4 * t4b5 - t4b5 * t4b4) - t4a4 + t4a5;
rel1(t4b2,t4b3) + _q^-1 * t4b3;
rel1(t4b3,t4b2) + _q^-1 * t4b2;
rel1(t4b4,t4b3);
rel1(t4b3,t4b4) + _q^-1 * t4b4;
t4b2*t4b4 - t4b4*t4b2;
t4b2*t4b5 - t4b5*t4b2;
rel1(t4b5,t4b3);
rel1(t4b3,t4b5) + _q^-1 * t4b5;  
########### t4 is the inverse of t4m ##############
_q^-1 * qcom2(qcom2(t4mb3,t4mb4),t4mb5)+ t4mb3*t4ma4 - b3;
_q * t4ma5 * t4mb5 - b4;
_q * t4ma4 * t4mb4 - b5;
_q^-1 * qcom2(t4b4,qcom2(t4b5,t4b3))+t4b3*t4a4 - b3;
_q^-1 * t4a4* t4b5 - b4;
_q^-1 * t4a5 * t4b4 - b5;
############ 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: #################################
t3b2:=qcom2(b2,b3);;
t3b3:=b3;;
t3b4:=qcom2(b4,b3);; 
t3b5:=qcom2(b5,b3);; 
t3a4:=a4;;
t3a5:=a5;;
########### and recall the definition of t4 ###############################
t4b2:=b2;;
t4b3:= _q^-1 * qcom2(qcom2(b3,b4),b5)+ b3*a4;;
t4b4:= _q * a5 * b5;;
t4b5:= _q * a4 * b4;;
t4a4:= a5;;
t4a5:= a4;;
##############################################################################
#### Observe, that both t3 and t4 act as the identity on  b1.             ####
#### Hence we only need to consider the action on the smaller subalgebra  ####
#### generated by the other generators.                                   ####
##############################################################################
t43b2:=qcom2(t4b2,t4b3);;
t43b3:=t4b3;;
t43b4:=qcom2(t4b4,t4b3);; 
t43b5:=qcom2(t4b5,t4b3);; 
t43a4:=t4a4;;
t43a5:=t4a5;;
###########
t34b2:=t3b2;;
t34b3:= _q^-1 * qcom2(qcom2(t3b3,t3b4),t3b5)+ t3b3*t3a4;;
t34b4:= _q * t3a5 * t3b5;;
t34b5:= _q * t3a4 * t3b4;;
t34a4:= t3a5;;
t34a5:= t3a4;;
##########
t343b2:=qcom2(t34b2,t34b3);;
t343b3:=t34b3;;
t343b4:=qcom2(t34b4,t34b3);; 
t343b5:=qcom2(t34b5,t34b3);;
t343a4:=t34a4;;
t343a5:=t34a5;;
###########
t434b2:=t43b2;;
t434b3:= _q^-1 * qcom2(qcom2(t43b3,t43b4),t43b5)+ t43b3*t43a4;;
t434b4:= _q * t43a5 * t43b5;;
t434b5:= _q * t43a4 * t43b4;;
t434a4:= t43a5;;
t434a5:= t43a4;;
##########
t4343b2:=qcom2(t434b2,t434b3);;
t4343b3:=t434b3;;
t4343b4:=qcom2(t434b4,t434b3);; 
t4343b5:=qcom2(t434b5,t434b3);; 
t4343a4:=t434a4;;
t4343a5:=t434a5;;
###########
t3434b2:=t343b2;;
t3434b3:= _q^-1 * qcom2(qcom2(t343b3,t343b4),t343b5)+ t343b3*t343a4;;
t3434b4:= _q * t343a5 * t343b5;;
t3434b5:= _q * t343a4 * t343b4;;
t3434a4:= t343a5;;
t3434a5:= t343a4;;
####################################################################
############ final verification of the type B braid relation: ######
############ all the followsing computations need to give 0   ######
####################################################################
t3434b2 - t4343b2; 
t3434b3 - t4343b3;
t3434b4 - t4343b4;
t3434b5 - t4343b5;
t3434a4 - t4343a4;
t3434a5 - t4343a5;
####################################################################
#### Now we approach the type A braid relation between t2 and #####
#### t3 in the same way, again writing t232b1 etc.            #####
####################################################################
t2b1:=qcom2(b1,b2);;
t2b2:=b2;;
t2b3:=qcom2(b3,b2);; 
t2b4:=b4;;
t2b5:=b5;;
t2a4:=a4;;
t2a5:=a5;;
#########################
t3b1:=b1;;
t3b2:=qcom2(b2,b3);;
t3b3:=b3;;
t3b4:=qcom2(b4,b3);; 
t3b5:=qcom2(b5,b3);; 
t3a4:=a4;;
t3a5:=a5;;
#########################
t32b1:=qcom2(t3b1,t3b2);;
t32b2:=t3b2;;
t32b3:=qcom2(t3b3,t3b2);; 
t32b4:=t3b4;;
t32b5:=t3b5;;
t32a4:=t3a4;;
t32a5:=t3a5;;
#########################
t23b1:=t2b1;;
t23b2:=qcom2(t2b2,t2b3);;
t23b3:=t2b3;;
t23b4:=qcom2(t2b4,t2b3);; 
t23b5:=qcom2(t2b5,t2b3);; 
t23a4:=t2a4;;
t23a5:=t2a5;;
#########################
t232b1:=qcom2(t23b1,t23b2);;
t232b2:=t23b2;;
t232b3:=qcom2(t23b3,t23b2);; 
t232b4:=t23b4;;
t232b5:=t23b5;; 
t232a4:=t23a4;;
t232a5:=t23a5;;
#########################
t323b1:=t32b1;;
t323b2:=qcom2(t32b2,t32b3);;
t323b3:=t32b3;;
t323b4:=qcom2(t32b4,t32b3);; 
t323b5:=qcom2(t32b5,t32b3);; 
t323a4:=t32a4;;
t323a5:=t32a5;;
###############################################################################
########## Final step in the verification of the type A braid relation ########
###############################################################################
t323b1 - t232b1;
t323b2 - t232b2;
t323b3 - t232b3;
t323b4 - t232b4;
t323b5 - t232b5;
t323a4 - t232a4;
t323a5 - t232a5;
###############################################################################

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