R:= RootSystem( "C",3);
 U:=QuantizedUEA(R); 
 g:= GeneratorsOfAlgebra( U ); 
 PositiveRoots(R); 
 PositiveRootsInConvexOrder(R); 
########## generators of U ###############
 f1:=g[1]; f2:=g[3]; f3:=g[9];
 e1:=g[16]; e2:=g[18]; e3:=g[24];
 k1:=g[10]; k1m:=g[11]; k2:=g[12]; k2m:=g[13]; k3:=g[14]; k3m:=g[15];
########## generators of Uq'(k) ###############
 b1:=f1-k1m*e1;
 b2:=f2-k2m*e2;
 b3:=f3-k3m*e3;
########## q-numbers #################### 
 2q:=_q+_q^-1;
 3q:=_q^2+1+_q^-2;
######### relations in Uq'(k) as functions #########
 rel0:=function(a,b)
   return a*b - b*a;
 end;
 rel1:=function(a,b)
   return a^2*b - (_q+_q^-1)*a*b*a + b*a^2 +_q^-1 *b;
 end;
 rel12:=function(a,b)
   return a^2*b - (_q^2+_q^-2)*a*b*a + b*a^2 +_q^-2 *b;
 end;
 rel2:=function(a,b)
   return b^3*a - 3q *b^2*a*b + 3q*  b*a*b^2 - a*b^3 +_q^-1*2q^2 *(b*a-a*b);
 end;
###############################################################################
########### We check that the generators bi really satisfy the relations ######
########### given Proposition 3.1                                        ######
###############################################################################
 rel0(b1,b3);
 rel1(b1,b2);
 rel1(b2,b1);
 rel12(b3,b2);
 rel2(b3,b2);
######## Definition of \tau_1^- ##########
 tau1mb1 := b1;;
 tau1mb2 := b1*b2 - _q *b2*b1;;
 tau1mb3 := b3;;
####### \tau_1^- is an algebra homomorphism #####
 rel0(tau1mb1,tau1mb3);
 rel1(tau1mb1,tau1mb2);
 rel1(tau1mb2,tau1mb1);
 rel12(tau1mb3,tau1mb2);
 rel2(tau1mb3,tau1mb2);
######## Definition of \tau_1 ##########
 tau1b1 := b1;;
 tau1b2 := b2*b1 - _q *b1*b2;;
 tau1b3 := b3;;
####### \tau_1 is an algebra homomorphism #####
 rel0(tau1b1,tau1b3);
 rel1(tau1b1,tau1b2);
 rel1(tau1b2,tau1b1);
 rel12(tau1b3,tau1b2);
 rel2(tau1b3,tau1b2);
###### \tau_1 is the inverse of \tau_1^- #####
  tau1b1*tau1b2 - _q *tau1b2*tau1b1 - b2;
  tau1mb2*tau1mb1 - _q *tau1mb1*tau1mb2 -b2;
######## Definition of \tau_2^- ##########
 tau2mb1 := b2*b1-_q * b1*b2;;
 tau2mb2 := b2;;
 tau2mb3 := 2q^-1*b2^2*b3-_q*b2*b3*b2+_q^2*2q^-1*b3*b2^2+  b3;;
####### \tau_2^- is an algebra homomorphism #####
 rel0(tau2mb1,tau2mb3);
 rel1(tau2mb1,tau2mb2);
 rel1(tau2mb2,tau2mb1);
 rel12(tau2mb3,tau2mb2);
 rel2(tau2mb3,tau2mb2);
######## Definition of \tau_2 ##########
 tau2b1 := b1*b2-_q * b2*b1;;
 tau2b2 := b2;;
 tau2b3 := 2q^-1*b3*b2^2 -_q*b2*b3*b2+_q^2*2q^-1*b2^2*b3 +  b3;;
####### \tau_2 is an algebra homomorphism #####
 rel0(tau2b1,tau2b3);
 rel1(tau2b1,tau2b2);
 rel1(tau2b2,tau2b1);
 rel12(tau2b3,tau2b2);
 rel2(tau2b3,tau2b2);
####### \tau_2 is the inverse of \tau_2^- #########
 tau2b2*tau2b1-_q * tau2b1*tau2b2 -b1;
 2q^-1*tau2b2^2*tau2b3-_q*tau2b2*tau2b3*tau2b2+_q^2*2q^-1*tau2b3*tau2b2^2+  tau2b3 -b3;
 tau2mb1*tau2mb2-_q * tau2mb2*tau2mb1 -b1;
 2q^-1*tau2mb3*tau2mb2^2 -_q*tau2mb2*tau2mb3*tau2mb2+_q^2*2q^-1*tau2mb2^2*tau2mb3 + tau2mb3-b3;
######## Definition of \tau_3^- ########################## 
 tau3mb1 := b1;;
 tau3mb2 := b3 * b2 - _q^2 * b2 * b3;;
 tau3mb3 := b3;;
####### \tau_3^- is an algebra homomorphism #####
 rel0(tau3mb1,tau3mb3);
 rel1(tau3mb1,tau3mb2);
 rel1(tau3mb2,tau3mb1);
 rel12(tau3mb3,tau3mb2);
 rel2(tau3mb3,tau3mb2);
######## Definition of \tau_3 ########################## 
 tau3b1 := b1;;
 tau3b2 := b2 * b3 - _q^2 * b3 * b2;;
 tau3b3 := b3;;
####### \tau_3^- is an algebra homomorphism #####
 rel0(tau3mb1,tau3mb3);
 rel1(tau3mb1,tau3mb2);
 rel1(tau3mb2,tau3mb1);
 rel12(tau3mb3,tau3mb2);
 rel2(tau3mb3,tau3mb2);
####### \tau_3^- is the inverse of \tau_3 ##########
 tau3b3 * tau3b2 - _q^2 * tau3b2 * tau3b3 - b2;
 tau3mb2 * tau3mb3 - _q^2 * tau3mb3 * tau3mb2 -b2;
##################################################################
######## Verifying the braid relations ###########################
######## We write t2323 for tau_2 tau_3 tau_2(b3) etc. ###########
##################################################################
t11:= b1;;
t12:= b2*b1 - _q *b1*b2;;
t13:= b3;;
t21:= b1*b2-_q * b2*b1;;
t22:= b2;;
t23:= 2q^-1*b3*b2^2 -_q*b2*b3*b2+_q^2*2q^-1*b2^2*b3 +  b3;;
t31:= b1;;
t32:= b2 * b3 - _q^2 * b3 * b2;;
t33:= b3;;
############
t311:= b1;;
t312:= t32*t31 - _q *t31*t32;;
t313:= b3;;
t131:= b1;;
t132:= t12 * t13 - _q^2 * t13 * t12;;
t133:= b3;;
############
t121:= t11*t12-_q * t12*t11;;
t122:= t12;;
t123:= 2q^-1*t13*t12^2 -_q*t12*t13*t12+_q^2*2q^-1*t12^2*t13 +  t13;;
##
t211:= t21;;
t212:= t22*t21 - _q *t21*t22;;
t213:= t23;;
##
t231:=t21;;
t232:= t22 * t23 - _q^2 * t23 * t22;;
t233:=t23;;
##
t321:=t31*t32-_q * t32*t31;;
t322:=t32;;
t323:=2q^-1*t33*t32^2 -_q*t32*t33*t32+_q^2*2q^-1*t32^2*t33 +  t33;;
############
t2121:= t211*t212-_q * t212*t211;;
t2122:= t212;;
t2123:= 2q^-1*t213*t212^2 -_q*t212*t213*t212+_q^2*2q^-1*t212^2*t213 +  t213;;
##
t1211:= t121;;
t1212:= t122*t121 - _q *t121*t122;;
t1213:= t123;;
##
t3231:=t321;;
t3232:=t322 * t323 - _q^2 * t323 * t322;;
t3233:=t323;;
##
t2321:=t231*t232-_q * t232*t231;;
t2322:=t232;;
t2323:=2q^-1*t233*t232^2 -_q*t232*t233*t232+_q^2*2q^-1*t232^2*t233 +  t233;;
############
t23231:=t2321;;
t23232:=t2322 * t2323 - _q^2 * t2323 * t2322;;
t23233:=t2323;;
##
t32321:=t3231*t3232-_q * t3232*t3231;;
t32322:=t3232;;
t32323:=2q^-1*t3233*t3232^2 -_q*t3232*t3233*t3232+_q^2*2q^-1*t3232^2*t3233 +  t3233;;
######## braid relations #################################
t132-t312;
t2121-t1211;
t2122-t1212;
t2123-t1213;
t32321-t23231;
t32322-t23232;
t32323-t23233;
######## :-) End :-) #####################################