tensorRules={t[A,A]->A+B+C+M,t[A,B]->A+2 B+C,t[A,C]->A+B,t[A,D]->2 D+E+F+Xb,t[A,E]->D+E,t[A,F]->D+F,t[A,M]->A,t[A,Xb]->D+Xb,t[B,A]->A+2 B+C,t[B,B]->2 A+2 B+C+M,t[B,C]->A+B+C,t[B,D]->3 D+E+F+Xb,t[B,E]->D+F+Xb,t[B,F]->D+E+Xb,t[B,M]->B,t[B,Xb]->D+E+F,t[C,A]->A+B,t[C,B]->A+B+C,t[C,C]->B+M,t[C,D]->D+E+F+Xb,t[C,E]->D,t[C,F]->D,t[C,M]->C,t[C,Xb]->D,t[D,N]->D,t[D,P]->2 D+E+F+Xb,t[D,Q]->2 D+E+F+Xb,t[D,S]->D,t[D,S2]->D,t[D,V2]->2 D+E+F+Xb,t[D,V3]->2 A+3 B+C+M,t[D,W]->A+B+C,t[D,X]->A+B+C,t[D,Z]->A+B+C,t[E,N]->E,t[E,P]->D+E,t[E,Q]->D+Xb,t[E,S]->F,t[E,S2]->Xb,t[E,V2]->D+F,t[E,V3]->A+B+C,t[E,W]->B,t[E,X]->B,t[E,Z]->A+M,t[F,N]->F,t[F,P]->D+Xb,t[F,Q]->D+F,t[F,S]->Xb,t[F,S2]->E,t[F,V2]->D+E,t[F,V3]->A+B+C,t[F,W]->A+M,t[F,X]->B,t[F,Z]->B,t[M,A]->A,t[M,B]->B,t[M,C]->C,t[M,D]->D,t[M,E]->E,t[M,F]->F,t[M,M]->M,t[M,Xb]->Xb,t[N,N]->N,t[N,P]->P,t[N,Q]->Q,t[N,S]->S,t[N,S2]->S2,t[N,V2]->V2,t[N,V3]->V3,t[N,W]->W,t[N,X]->X,t[N,Z]->Z,t[P,N]->P,t[P,P]->N+P+Q+V2,t[P,Q]->P+Q+S2+V2,t[P,S]->V2,t[P,S2]->Q,t[P,V2]->P+Q+S+V2,t[P,V3]->2 V3+W+X+Z,t[P,W]->V3+X,t[P,X]->V3+W,t[P,Z]->V3+Z,t[Q,N]->Q,t[Q,P]->P+Q+S+V2,t[Q,Q]->N+P+Q+V2,t[Q,S]->P,t[Q,S2]->V2,t[Q,V2]->P+Q+S2+V2,t[Q,V3]->2 V3+W+X+Z,t[Q,W]->V3+W,t[Q,X]->V3+Z,t[Q,Z]->V3+X,t[S,N]->S,t[S,P]->Q,t[S,Q]->V2,t[S,S]->S2,t[S,S2]->N,t[S,V2]->P,t[S,V3]->V3,t[S,W]->Z,t[S,X]->W,t[S,Z]->X,t[S2,N]->S2,t[S2,P]->V2,t[S2,Q]->P,t[S2,S]->N,t[S2,S2]->S,t[S2,V2]->Q,t[S2,V3]->V3,t[S2,W]->X,t[S2,X]->Z,t[S2,Z]->W,t[V2,N]->V2,t[V2,P]->P+Q+S2+V2,t[V2,Q]->P+Q+S+V2,t[V2,S]->Q,t[V2,S2]->P,t[V2,V2]->N+P+Q+V2,t[V2,V3]->2 V3+W+X+Z,t[V2,W]->V3+Z,t[V2,X]->V3+X,t[V2,Z]->V3+W,t[V3,A]->2 V3+W+X+Z,t[V3,B]->3 V3+W+X+Z,t[V3,C]->V3+W+X+Z,t[V3,D]->N+2 P+2 Q+S+S2+2 V2,t[V3,E]->P+Q+V2,t[V3,F]->P+Q+V2,t[V3,M]->V3,t[V3,Xb]->P+Q+V2,t[W,A]->V3+W,t[W,B]->V3+X+Z,t[W,C]->V3,t[W,D]->P+Q+V2,t[W,E]->S2+V2,t[W,F]->N+Q,t[W,M]->W,t[W,Xb]->P+S,t[X,A]->V3+X,t[X,B]->V3+W+Z,t[X,C]->V3,t[X,D]->P+Q+V2,t[X,E]->Q+S,t[X,F]->P+S2,t[X,M]->X,t[X,Xb]->N+V2,t[Xb,N]->Xb,t[Xb,P]->D+F,t[Xb,Q]->D+E,t[Xb,S]->E,t[Xb,S2]->F,t[Xb,V2]->D+Xb,t[Xb,V3]->A+B+C,t[Xb,W]->B,t[Xb,X]->A+M,t[Xb,Z]->B,t[Z,A]->V3+Z,t[Z,B]->V3+W+X,t[Z,C]->V3,t[Z,D]->P+Q+V2,t[Z,E]->N+P,t[Z,F]->S+V2,t[Z,M]->Z,t[Z,Xb]->Q+S2};

tensorLabellings={N,M};
bimodules[N,N]={N,V2,P,Q,S,S2};
bimodules[N,M]={X,V3,W,Z};
bimodules[M,N]={Xb,D,E,F};
bimodules[M,M]={M,A,B,C};

braidingRules=Cases[{Lp[X,D]->V3,Lp[V3,D]->V3,Lp[W,D]->V3,Lp[Z,D]->V3,Lp[Xb,V3]->D,Lp[D,V3]->D,Lp[E,V3]->D,Lp[F,V3]->D,Rp[D,Xb]->V3,Rp[D,D]->V3,Rp[D,E]->V3,Rp[D,F]->V3,Rp[V3,X]->D,Rp[V3,V3]->D,Rp[V3,W]->D,Rp[V3,Z]->D,Rp[N,N]->N,Rp[V2,N]->V2,Rp[P,N]->P,Rp[Q,N]->Q,Rp[S,N]->S,Rp[S2,N]->S2,Lp[N,N]->N,Lp[N,V2]->V2,Lp[N,P]->P,Lp[N,Q]->Q,Lp[N,S]->S,Lp[N,S2]->S2,Rp[M,M]->M,Rp[A,M]->A,Rp[B,M]->B,Rp[C,M]->C,Lp[M,M]->M,Lp[M,A]->A,Lp[M,B]->B,Lp[M,C]->C,Lp[N,N]->N,Lp[V2,N]->N,Lp[P,N]->N,Lp[Q,N]->N,Lp[S,N]->N,Lp[S2,N]->N,Rp[N,N]->N,Rp[N,V2]->N,Rp[N,P]->N,Rp[N,Q]->N,Rp[N,S]->N,Rp[N,S2]->N,Lp[M,M]->M,Lp[A,M]->M,Lp[B,M]->M,Lp[C,M]->M,Rp[M,M]->M,Rp[M,A]->M,Rp[M,B]->M,Rp[M,C]->M,Lp[V2,V2]->P,Rp[V2,V2]->P,Lp[V2,P]->P,Rp[V2,P]->Q,Lp[V2,Q]->P,Rp[V2,Q]->V2,Lp[V2,S]->N,Rp[V2,S]->Q,Lp[V2,S2]->N,Rp[V2,S2]->P,Lp[P,V2]->P,Rp[P,V2]->V2,Lp[P,P]->P,Rp[P,P]->P,Lp[P,Q]->P,Rp[P,Q]->Q,Lp[P,S]->N,Rp[P,S]->V2,Lp[P,S2]->N,Rp[P,S2]->Q,Lp[Q,V2]->P,Rp[Q,V2]->Q,Lp[Q,P]->P,Rp[Q,P]->V2,Lp[Q,Q]->P,Rp[Q,Q]->P,Lp[Q,S]->N,Rp[Q,S]->P,Lp[Q,S2]->N,Rp[Q,S2]->V2,Lp[S,V2]->V2,Rp[S,V2]->S2,Lp[S,P]->P,Rp[S,P]->S2,Lp[S,Q]->Q,Rp[S,Q]->S2,Lp[S,S]->S,Rp[S,S]->S,Lp[S,S2]->S2,Rp[S,S2]->S,Lp[S2,V2]->V2,Rp[S2,V2]->S,Lp[S2,P]->P,Rp[S2,P]->S,Lp[S2,Q]->Q,Rp[S2,Q]->S,Lp[S2,S]->S,Rp[S2,S]->S2,Lp[S2,S2]->S2,Rp[S2,S2]->S2},(_[A_,B_]->C_)/;MemberQ[bimodules[N,N],A]∧MemberQ[bimodules[N,N],B]∧MemberQ[bimodules[N,N],C]];

(Outer[List,bimodules[N,N],bimodules[N,N]])/.{A_,B_}:>{{Lp[A,B],Rp[A,B]},{A,B}}/.braidingRules//TableForm

quasiTriangularityEquation[a_,b_,c_]:={t[Rp[a,Lp[b,c]],Rp[b,c]]==Rp[t[a,b],c],Lp[t[a,b],c]==Lp[a,Lp[b,c]]}

quasiTriangularityEquations=Flatten[quasiTriangularityEquation@@#&/@Flatten[Outer[List,bimodules[N,N],bimodules[N,N],bimodules[N,N]],2]];

Lp[Y_,X_Plus]:=Lp[Y,#]&/@X
Rp[Y_,X_Plus]:=Rp[Y,#]&/@X
Lp[X_Plus,Y_]:=Lp[#,Y]&/@X
Rp[X_Plus,Y_]:=Rp[#,Y]&/@X

Simplify[quasiTriangularityEquations//.tensorRules~Join~braidingRules]

(3^27)

braidingRules1=Cases[{Lp[A,M]->M,Lp[B,M]->M,Lp[C,M]->M,Lp[D,V3]->D,Lp[E,V3]->D,Lp[F,V3]->D,Lp[M,A]->A,Lp[M,B]->B,Lp[M,C]->C,Lp[M,M]->M,Lp[N,N]->N,Lp[N,P]->P,Lp[N,Q]->Q,Lp[N,S]->S,Lp[N,S2]->S2,Lp[N,V2]->V2,Lp[P,N]->N,Lp[Q,N]->N,Lp[S,N]->N,Lp[S,S]->S,Lp[S,S2]->S2,Lp[S2,N]->N,Lp[S2,S]->S,Lp[S2,S2]->S2,Lp[V2,N]->N,Lp[V3,D]->V3,Lp[W,D]->V3,Lp[X,D]->V3,Lp[Xb,V3]->D,Lp[Z,D]->V3,Rp[A,M]->A,Rp[B,M]->B,Rp[C,M]->C,Rp[D,D]->V3,Rp[D,E]->V3,Rp[D,F]->V3,Rp[D,Xb]->V3,Rp[M,A]->M,Rp[M,B]->M,Rp[M,C]->M,Rp[M,M]->M,Rp[N,N]->N,Rp[N,P]->N,Rp[N,Q]->N,Rp[N,S]->N,Rp[N,S2]->N,Rp[N,V2]->N,Rp[P,N]->P,Rp[Q,N]->Q,Rp[S,N]->S,Rp[S,S]->S,Rp[S,S2]->S,Rp[S2,N]->S2,Rp[S2,S]->S2,Rp[S2,S2]->S2,Rp[V2,N]->V2,Rp[V3,V3]->D,Rp[V3,W]->D,Rp[V3,X]->D,Rp[V3,Z]->D,t[A,A]->A+B+C+M,t[A,B]->A+2 B+C,t[A,C]->A+B,t[A,D]->2 D+E+F+Xb,t[A,E]->D+E,t[A,F]->D+F,t[A,M]->A,t[A,Xb]->D+Xb,t[B,A]->A+2 B+C,t[B,B]->2 A+2 B+C+M,t[B,C]->A+B+C,t[B,D]->3 D+E+F+Xb,t[B,E]->D+F+Xb,t[B,F]->D+E+Xb,t[B,M]->B,t[B,Xb]->D+E+F,t[C,A]->A+B,t[C,B]->A+B+C,t[C,C]->B+M,t[C,D]->D+E+F+Xb,t[C,E]->D,t[C,F]->D,t[C,M]->C,t[C,Xb]->D,t[D,N]->D,t[D,P]->2 D+E+F+Xb,t[D,Q]->2 D+E+F+Xb,t[D,S]->D,t[D,S2]->D,t[D,V2]->2 D+E+F+Xb,t[D,V3]->2 A+3 B+C+M,t[D,W]->A+B+C,t[D,X]->A+B+C,t[D,Z]->A+B+C,t[E,N]->E,t[E,P]->D+E,t[E,Q]->D+Xb,t[E,S]->F,t[E,S2]->Xb,t[E,V2]->D+F,t[E,V3]->A+B+C,t[E,W]->B,t[E,X]->B,t[E,Z]->A+M,t[F,N]->F,t[F,P]->D+Xb,t[F,Q]->D+F,t[F,S]->Xb,t[F,S2]->E,t[F,V2]->D+E,t[F,V3]->A+B+C,t[F,W]->A+M,t[F,X]->B,t[F,Z]->B,t[M,A]->A,t[M,B]->B,t[M,C]->C,t[M,D]->D,t[M,E]->E,t[M,F]->F,t[M,M]->M,t[M,Xb]->Xb,t[N,N]->N,t[N,P]->P,t[N,Q]->Q,t[N,S]->S,t[N,S2]->S2,t[N,V2]->V2,t[N,V3]->V3,t[N,W]->W,t[N,X]->X,t[N,Z]->Z,t[P,N]->P,t[P,P]->N+P+Q+V2,t[P,Q]->P+Q+S2+V2,t[P,S]->V2,t[P,S2]->Q,t[P,V2]->P+Q+S+V2,t[P,V3]->2 V3+W+X+Z,t[P,W]->V3+X,t[P,X]->V3+W,t[P,Z]->V3+Z,t[Q,N]->Q,t[Q,P]->P+Q+S+V2,t[Q,Q]->N+P+Q+V2,t[Q,S]->P,t[Q,S2]->V2,t[Q,V2]->P+Q+S2+V2,t[Q,V3]->2 V3+W+X+Z,t[Q,W]->V3+W,t[Q,X]->V3+Z,t[Q,Z]->V3+X,t[S,N]->S,t[S,P]->Q,t[S,Q]->V2,t[S,S]->S2,t[S,S2]->N,t[S,V2]->P,t[S,V3]->V3,t[S,W]->Z,t[S,X]->W,t[S,Z]->X,t[S2,N]->S2,t[S2,P]->V2,t[S2,Q]->P,t[S2,S]->N,t[S2,S2]->S,t[S2,V2]->Q,t[S2,V3]->V3,t[S2,W]->X,t[S2,X]->Z,t[S2,Z]->W,t[V2,N]->V2,t[V2,P]->P+Q+S2+V2,t[V2,Q]->P+Q+S+V2,t[V2,S]->Q,t[V2,S2]->P,t[V2,V2]->N+P+Q+V2,t[V2,V3]->2 V3+W+X+Z,t[V2,W]->V3+Z,t[V2,X]->V3+X,t[V2,Z]->V3+W,t[V3,A]->2 V3+W+X+Z,t[V3,B]->3 V3+W+X+Z,t[V3,C]->V3+W+X+Z,t[V3,D]->N+2 P+2 Q+S+S2+2 V2,t[V3,E]->P+Q+V2,t[V3,F]->P+Q+V2,t[V3,M]->V3,t[V3,Xb]->P+Q+V2,t[W,A]->V3+W,t[W,B]->V3+X+Z,t[W,C]->V3,t[W,D]->P+Q+V2,t[W,E]->S2+V2,t[W,F]->N+Q,t[W,M]->W,t[W,Xb]->P+S,t[X,A]->V3+X,t[X,B]->V3+W+Z,t[X,C]->V3,t[X,D]->P+Q+V2,t[X,E]->Q+S,t[X,F]->P+S2,t[X,M]->X,t[X,Xb]->N+V2,t[Xb,N]->Xb,t[Xb,P]->D+F,t[Xb,Q]->D+E,t[Xb,S]->E,t[Xb,S2]->F,t[Xb,V2]->D+Xb,t[Xb,V3]->A+B+C,t[Xb,W]->B,t[Xb,X]->A+M,t[Xb,Z]->B,t[Z,A]->V3+Z,t[Z,B]->V3+W+X,t[Z,C]->V3,t[Z,D]->P+Q+V2,t[Z,E]->N+P,t[Z,F]->S+V2,t[Z,M]->Z,t[Z,Xb]->Q+S2,Lp[V2,V2]->P,Rp[V2,V2]->P,Lp[V2,P]->V2,Rp[V2,P]->P,Lp[V2,Q]->Q,Rp[V2,Q]->P,Lp[V2,S]->S,Rp[V2,S]->P,Lp[V2,S2]->S2,Rp[V2,S2]->Q,Lp[P,V2]->V2,Rp[P,V2]->Q,Lp[P,P]->Q,Rp[P,P]->Q,Lp[P,Q]->P,Rp[P,Q]->Q,Lp[P,S]->S,Rp[P,S]->Q,Lp[P,S2]->S2,Rp[P,S2]->V2,Lp[Q,V2]->Q,Rp[Q,V2]->V2,Lp[Q,P]->P,Rp[Q,P]->V2,Lp[Q,Q]->V2,Rp[Q,Q]->V2,Lp[Q,S]->S,Rp[Q,S]->V2,Lp[Q,S2]->S2,Rp[Q,S2]->P,Lp[S,V2]->Q,Rp[S,V2]->S,Lp[S,P]->V2,Rp[S,P]->S,Lp[S,Q]->P,Rp[S,Q]->S,Lp[S2,V2]->P,Rp[S2,V2]->S2,Lp[S2,P]->Q,Rp[S2,P]->S2,Lp[S2,Q]->V2,Rp[S2,Q]->S2},(_[A_,B_]->C_)/;MemberQ[bimodules[N,N],A]∧MemberQ[bimodules[N,N],B]∧MemberQ[bimodules[N,N],C]]

braidingRules2=Cases[{Lp[A,M]->M,Lp[B,M]->M,Lp[C,M]->M,Lp[D,V3]->D,Lp[E,V3]->D,Lp[F,V3]->D,Lp[M,A]->A,Lp[M,B]->B,Lp[M,C]->C,Lp[M,M]->M,Lp[N,N]->N,Lp[N,P]->P,Lp[N,Q]->Q,Lp[N,S]->S,Lp[N,S2]->S2,Lp[N,V2]->V2,Lp[P,N]->N,Lp[Q,N]->N,Lp[S,N]->N,Lp[S,S]->S,Lp[S,S2]->S2,Lp[S2,N]->N,Lp[S2,S]->S,Lp[S2,S2]->S2,Lp[V2,N]->N,Lp[V3,D]->V3,Lp[W,D]->V3,Lp[X,D]->V3,Lp[Xb,V3]->D,Lp[Z,D]->V3,Rp[A,M]->A,Rp[B,M]->B,Rp[C,M]->C,Rp[D,D]->V3,Rp[D,E]->V3,Rp[D,F]->V3,Rp[D,Xb]->V3,Rp[M,A]->M,Rp[M,B]->M,Rp[M,C]->M,Rp[M,M]->M,Rp[N,N]->N,Rp[N,P]->N,Rp[N,Q]->N,Rp[N,S]->N,Rp[N,S2]->N,Rp[N,V2]->N,Rp[P,N]->P,Rp[Q,N]->Q,Rp[S,N]->S,Rp[S,S]->S,Rp[S,S2]->S,Rp[S2,N]->S2,Rp[S2,S]->S2,Rp[S2,S2]->S2,Rp[V2,N]->V2,Rp[V3,V3]->D,Rp[V3,W]->D,Rp[V3,X]->D,Rp[V3,Z]->D,t[A,A]->A+B+C+M,t[A,B]->A+2 B+C,t[A,C]->A+B,t[A,D]->2 D+E+F+Xb,t[A,E]->D+E,t[A,F]->D+F,t[A,M]->A,t[A,Xb]->D+Xb,t[B,A]->A+2 B+C,t[B,B]->2 A+2 B+C+M,t[B,C]->A+B+C,t[B,D]->3 D+E+F+Xb,t[B,E]->D+F+Xb,t[B,F]->D+E+Xb,t[B,M]->B,t[B,Xb]->D+E+F,t[C,A]->A+B,t[C,B]->A+B+C,t[C,C]->B+M,t[C,D]->D+E+F+Xb,t[C,E]->D,t[C,F]->D,t[C,M]->C,t[C,Xb]->D,t[D,N]->D,t[D,P]->2 D+E+F+Xb,t[D,Q]->2 D+E+F+Xb,t[D,S]->D,t[D,S2]->D,t[D,V2]->2 D+E+F+Xb,t[D,V3]->2 A+3 B+C+M,t[D,W]->A+B+C,t[D,X]->A+B+C,t[D,Z]->A+B+C,t[E,N]->E,t[E,P]->D+E,t[E,Q]->D+Xb,t[E,S]->F,t[E,S2]->Xb,t[E,V2]->D+F,t[E,V3]->A+B+C,t[E,W]->B,t[E,X]->B,t[E,Z]->A+M,t[F,N]->F,t[F,P]->D+Xb,t[F,Q]->D+F,t[F,S]->Xb,t[F,S2]->E,t[F,V2]->D+E,t[F,V3]->A+B+C,t[F,W]->A+M,t[F,X]->B,t[F,Z]->B,t[M,A]->A,t[M,B]->B,t[M,C]->C,t[M,D]->D,t[M,E]->E,t[M,F]->F,t[M,M]->M,t[M,Xb]->Xb,t[N,N]->N,t[N,P]->P,t[N,Q]->Q,t[N,S]->S,t[N,S2]->S2,t[N,V2]->V2,t[N,V3]->V3,t[N,W]->W,t[N,X]->X,t[N,Z]->Z,t[P,N]->P,t[P,P]->N+P+Q+V2,t[P,Q]->P+Q+S2+V2,t[P,S]->V2,t[P,S2]->Q,t[P,V2]->P+Q+S+V2,t[P,V3]->2 V3+W+X+Z,t[P,W]->V3+X,t[P,X]->V3+W,t[P,Z]->V3+Z,t[Q,N]->Q,t[Q,P]->P+Q+S+V2,t[Q,Q]->N+P+Q+V2,t[Q,S]->P,t[Q,S2]->V2,t[Q,V2]->P+Q+S2+V2,t[Q,V3]->2 V3+W+X+Z,t[Q,W]->V3+W,t[Q,X]->V3+Z,t[Q,Z]->V3+X,t[S,N]->S,t[S,P]->Q,t[S,Q]->V2,t[S,S]->S2,t[S,S2]->N,t[S,V2]->P,t[S,V3]->V3,t[S,W]->Z,t[S,X]->W,t[S,Z]->X,t[S2,N]->S2,t[S2,P]->V2,t[S2,Q]->P,t[S2,S]->N,t[S2,S2]->S,t[S2,V2]->Q,t[S2,V3]->V3,t[S2,W]->X,t[S2,X]->Z,t[S2,Z]->W,t[V2,N]->V2,t[V2,P]->P+Q+S2+V2,t[V2,Q]->P+Q+S+V2,t[V2,S]->Q,t[V2,S2]->P,t[V2,V2]->N+P+Q+V2,t[V2,V3]->2 V3+W+X+Z,t[V2,W]->V3+Z,t[V2,X]->V3+X,t[V2,Z]->V3+W,t[V3,A]->2 V3+W+X+Z,t[V3,B]->3 V3+W+X+Z,t[V3,C]->V3+W+X+Z,t[V3,D]->N+2 P+2 Q+S+S2+2 V2,t[V3,E]->P+Q+V2,t[V3,F]->P+Q+V2,t[V3,M]->V3,t[V3,Xb]->P+Q+V2,t[W,A]->V3+W,t[W,B]->V3+X+Z,t[W,C]->V3,t[W,D]->P+Q+V2,t[W,E]->S2+V2,t[W,F]->N+Q,t[W,M]->W,t[W,Xb]->P+S,t[X,A]->V3+X,t[X,B]->V3+W+Z,t[X,C]->V3,t[X,D]->P+Q+V2,t[X,E]->Q+S,t[X,F]->P+S2,t[X,M]->X,t[X,Xb]->N+V2,t[Xb,N]->Xb,t[Xb,P]->D+F,t[Xb,Q]->D+E,t[Xb,S]->E,t[Xb,S2]->F,t[Xb,V2]->D+Xb,t[Xb,V3]->A+B+C,t[Xb,W]->B,t[Xb,X]->A+M,t[Xb,Z]->B,t[Z,A]->V3+Z,t[Z,B]->V3+W+X,t[Z,C]->V3,t[Z,D]->P+Q+V2,t[Z,E]->N+P,t[Z,F]->S+V2,t[Z,M]->Z,t[Z,Xb]->Q+S2,Lp[V2,V2]->P,Rp[V2,V2]->P,Lp[V2,P]->V2,Rp[V2,P]->P,Lp[V2,Q]->Q,Rp[V2,Q]->P,Lp[V2,S]->S,Rp[V2,S]->P,Lp[V2,S2]->S2,Rp[V2,S2]->Q,Lp[P,V2]->V2,Rp[P,V2]->Q,Lp[P,P]->Q,Rp[P,P]->Q,Lp[P,Q]->P,Rp[P,Q]->Q,Lp[P,S]->S,Rp[P,S]->Q,Lp[P,S2]->S2,Rp[P,S2]->V2,Lp[Q,V2]->Q,Rp[Q,V2]->V2,Lp[Q,P]->P,Rp[Q,P]->V2,Lp[Q,Q]->V2,Rp[Q,Q]->V2,Lp[Q,S]->S,Rp[Q,S]->V2,Lp[Q,S2]->S2,Rp[Q,S2]->P,Lp[S,V2]->Q,Rp[S,V2]->S,Lp[S,P]->V2,Rp[S,P]->S,Lp[S,Q]->P,Rp[S,Q]->S,Lp[S2,V2]->P,Rp[S2,V2]->S2,Lp[S2,P]->Q,Rp[S2,P]->S2,Lp[S2,Q]->V2,Rp[S2,Q]->S2},(_[A_,B_]->C_)/;MemberQ[bimodules[N,N],A]∧MemberQ[bimodules[N,N],B]∧MemberQ[bimodules[N,N],C]]

braidingRules3=Cases[{Lp[A,M]->M,Lp[B,M]->M,Lp[C,M]->M,Lp[D,V3]->D,Lp[E,V3]->D,Lp[F,V3]->D,Lp[M,A]->A,Lp[M,B]->B,Lp[M,C]->C,Lp[M,M]->M,Lp[N,N]->N,Lp[N,P]->P,Lp[N,Q]->Q,Lp[N,S]->S,Lp[N,S2]->S2,Lp[N,V2]->V2,Lp[P,N]->N,Lp[Q,N]->N,Lp[S,N]->N,Lp[S,S]->S,Lp[S,S2]->S2,Lp[S2,N]->N,Lp[S2,S]->S,Lp[S2,S2]->S2,Lp[V2,N]->N,Lp[V3,D]->V3,Lp[W,D]->V3,Lp[X,D]->V3,Lp[Xb,V3]->D,Lp[Z,D]->V3,Rp[A,M]->A,Rp[B,M]->B,Rp[C,M]->C,Rp[D,D]->V3,Rp[D,E]->V3,Rp[D,F]->V3,Rp[D,Xb]->V3,Rp[M,A]->M,Rp[M,B]->M,Rp[M,C]->M,Rp[M,M]->M,Rp[N,N]->N,Rp[N,P]->N,Rp[N,Q]->N,Rp[N,S]->N,Rp[N,S2]->N,Rp[N,V2]->N,Rp[P,N]->P,Rp[Q,N]->Q,Rp[S,N]->S,Rp[S,S]->S,Rp[S,S2]->S,Rp[S2,N]->S2,Rp[S2,S]->S2,Rp[S2,S2]->S2,Rp[V2,N]->V2,Rp[V3,V3]->D,Rp[V3,W]->D,Rp[V3,X]->D,Rp[V3,Z]->D,t[A,A]->A+B+C+M,t[A,B]->A+2 B+C,t[A,C]->A+B,t[A,D]->2 D+E+F+Xb,t[A,E]->D+E,t[A,F]->D+F,t[A,M]->A,t[A,Xb]->D+Xb,t[B,A]->A+2 B+C,t[B,B]->2 A+2 B+C+M,t[B,C]->A+B+C,t[B,D]->3 D+E+F+Xb,t[B,E]->D+F+Xb,t[B,F]->D+E+Xb,t[B,M]->B,t[B,Xb]->D+E+F,t[C,A]->A+B,t[C,B]->A+B+C,t[C,C]->B+M,t[C,D]->D+E+F+Xb,t[C,E]->D,t[C,F]->D,t[C,M]->C,t[C,Xb]->D,t[D,N]->D,t[D,P]->2 D+E+F+Xb,t[D,Q]->2 D+E+F+Xb,t[D,S]->D,t[D,S2]->D,t[D,V2]->2 D+E+F+Xb,t[D,V3]->2 A+3 B+C+M,t[D,W]->A+B+C,t[D,X]->A+B+C,t[D,Z]->A+B+C,t[E,N]->E,t[E,P]->D+E,t[E,Q]->D+Xb,t[E,S]->F,t[E,S2]->Xb,t[E,V2]->D+F,t[E,V3]->A+B+C,t[E,W]->B,t[E,X]->B,t[E,Z]->A+M,t[F,N]->F,t[F,P]->D+Xb,t[F,Q]->D+F,t[F,S]->Xb,t[F,S2]->E,t[F,V2]->D+E,t[F,V3]->A+B+C,t[F,W]->A+M,t[F,X]->B,t[F,Z]->B,t[M,A]->A,t[M,B]->B,t[M,C]->C,t[M,D]->D,t[M,E]->E,t[M,F]->F,t[M,M]->M,t[M,Xb]->Xb,t[N,N]->N,t[N,P]->P,t[N,Q]->Q,t[N,S]->S,t[N,S2]->S2,t[N,V2]->V2,t[N,V3]->V3,t[N,W]->W,t[N,X]->X,t[N,Z]->Z,t[P,N]->P,t[P,P]->N+P+Q+V2,t[P,Q]->P+Q+S2+V2,t[P,S]->V2,t[P,S2]->Q,t[P,V2]->P+Q+S+V2,t[P,V3]->2 V3+W+X+Z,t[P,W]->V3+X,t[P,X]->V3+W,t[P,Z]->V3+Z,t[Q,N]->Q,t[Q,P]->P+Q+S+V2,t[Q,Q]->N+P+Q+V2,t[Q,S]->P,t[Q,S2]->V2,t[Q,V2]->P+Q+S2+V2,t[Q,V3]->2 V3+W+X+Z,t[Q,W]->V3+W,t[Q,X]->V3+Z,t[Q,Z]->V3+X,t[S,N]->S,t[S,P]->Q,t[S,Q]->V2,t[S,S]->S2,t[S,S2]->N,t[S,V2]->P,t[S,V3]->V3,t[S,W]->Z,t[S,X]->W,t[S,Z]->X,t[S2,N]->S2,t[S2,P]->V2,t[S2,Q]->P,t[S2,S]->N,t[S2,S2]->S,t[S2,V2]->Q,t[S2,V3]->V3,t[S2,W]->X,t[S2,X]->Z,t[S2,Z]->W,t[V2,N]->V2,t[V2,P]->P+Q+S2+V2,t[V2,Q]->P+Q+S+V2,t[V2,S]->Q,t[V2,S2]->P,t[V2,V2]->N+P+Q+V2,t[V2,V3]->2 V3+W+X+Z,t[V2,W]->V3+Z,t[V2,X]->V3+X,t[V2,Z]->V3+W,t[V3,A]->2 V3+W+X+Z,t[V3,B]->3 V3+W+X+Z,t[V3,C]->V3+W+X+Z,t[V3,D]->N+2 P+2 Q+S+S2+2 V2,t[V3,E]->P+Q+V2,t[V3,F]->P+Q+V2,t[V3,M]->V3,t[V3,Xb]->P+Q+V2,t[W,A]->V3+W,t[W,B]->V3+X+Z,t[W,C]->V3,t[W,D]->P+Q+V2,t[W,E]->S2+V2,t[W,F]->N+Q,t[W,M]->W,t[W,Xb]->P+S,t[X,A]->V3+X,t[X,B]->V3+W+Z,t[X,C]->V3,t[X,D]->P+Q+V2,t[X,E]->Q+S,t[X,F]->P+S2,t[X,M]->X,t[X,Xb]->N+V2,t[Xb,N]->Xb,t[Xb,P]->D+F,t[Xb,Q]->D+E,t[Xb,S]->E,t[Xb,S2]->F,t[Xb,V2]->D+Xb,t[Xb,V3]->A+B+C,t[Xb,W]->B,t[Xb,X]->A+M,t[Xb,Z]->B,t[Z,A]->V3+Z,t[Z,B]->V3+W+X,t[Z,C]->V3,t[Z,D]->P+Q+V2,t[Z,E]->N+P,t[Z,F]->S+V2,t[Z,M]->Z,t[Z,Xb]->Q+S2,Lp[V2,V2]->P,Rp[V2,V2]->P,Lp[V2,P]->V2,Rp[V2,P]->P,Lp[V2,Q]->Q,Rp[V2,Q]->P,Lp[V2,S]->S2,Rp[V2,S]->V2,Lp[V2,S2]->S,Rp[V2,S2]->V2,Lp[P,V2]->P,Rp[P,V2]->V2,Lp[P,P]->V2,Rp[P,P]->V2,Lp[P,Q]->Q,Rp[P,Q]->V2,Lp[P,S]->S2,Rp[P,S]->P,Lp[P,S2]->S,Rp[P,S2]->P,Lp[Q,V2]->P,Rp[Q,V2]->Q,Lp[Q,P]->V2,Rp[Q,P]->Q,Lp[Q,Q]->Q,Rp[Q,Q]->Q,Lp[Q,S]->S2,Rp[Q,S]->Q,Lp[Q,S2]->S,Rp[Q,S2]->Q,Lp[S,V2]->V2,Rp[S,V2]->S2,Lp[S,P]->P,Rp[S,P]->S2,Lp[S,Q]->Q,Rp[S,Q]->S2,Lp[S2,V2]->V2,Rp[S2,V2]->S,Lp[S2,P]->P,Rp[S2,P]->S,Lp[S2,Q]->Q,Rp[S2,Q]->S},(_[A_,B_]->C_)/;MemberQ[bimodules[N,N],A]∧MemberQ[bimodules[N,N],B]∧MemberQ[bimodules[N,N],C]]

braidingRules4=Cases[{Lp[A,M]->M,Lp[B,M]->M,Lp[C,M]->M,Lp[D,V3]->D,Lp[E,V3]->D,Lp[F,V3]->D,Lp[M,A]->A,Lp[M,B]->B,Lp[M,C]->C,Lp[M,M]->M,Lp[N,N]->N,Lp[N,P]->P,Lp[N,Q]->Q,Lp[N,S]->S,Lp[N,S2]->S2,Lp[N,V2]->V2,Lp[P,N]->N,Lp[Q,N]->N,Lp[S,N]->N,Lp[S,S]->S,Lp[S,S2]->S2,Lp[S2,N]->N,Lp[S2,S]->S,Lp[S2,S2]->S2,Lp[V2,N]->N,Lp[V3,D]->V3,Lp[W,D]->V3,Lp[X,D]->V3,Lp[Xb,V3]->D,Lp[Z,D]->V3,Rp[A,M]->A,Rp[B,M]->B,Rp[C,M]->C,Rp[D,D]->V3,Rp[D,E]->V3,Rp[D,F]->V3,Rp[D,Xb]->V3,Rp[M,A]->M,Rp[M,B]->M,Rp[M,C]->M,Rp[M,M]->M,Rp[N,N]->N,Rp[N,P]->N,Rp[N,Q]->N,Rp[N,S]->N,Rp[N,S2]->N,Rp[N,V2]->N,Rp[P,N]->P,Rp[Q,N]->Q,Rp[S,N]->S,Rp[S,S]->S,Rp[S,S2]->S,Rp[S2,N]->S2,Rp[S2,S]->S2,Rp[S2,S2]->S2,Rp[V2,N]->V2,Rp[V3,V3]->D,Rp[V3,W]->D,Rp[V3,X]->D,Rp[V3,Z]->D,t[A,A]->A+B+C+M,t[A,B]->A+2 B+C,t[A,C]->A+B,t[A,D]->2 D+E+F+Xb,t[A,E]->D+E,t[A,F]->D+F,t[A,M]->A,t[A,Xb]->D+Xb,t[B,A]->A+2 B+C,t[B,B]->2 A+2 B+C+M,t[B,C]->A+B+C,t[B,D]->3 D+E+F+Xb,t[B,E]->D+F+Xb,t[B,F]->D+E+Xb,t[B,M]->B,t[B,Xb]->D+E+F,t[C,A]->A+B,t[C,B]->A+B+C,t[C,C]->B+M,t[C,D]->D+E+F+Xb,t[C,E]->D,t[C,F]->D,t[C,M]->C,t[C,Xb]->D,t[D,N]->D,t[D,P]->2 D+E+F+Xb,t[D,Q]->2 D+E+F+Xb,t[D,S]->D,t[D,S2]->D,t[D,V2]->2 D+E+F+Xb,t[D,V3]->2 A+3 B+C+M,t[D,W]->A+B+C,t[D,X]->A+B+C,t[D,Z]->A+B+C,t[E,N]->E,t[E,P]->D+E,t[E,Q]->D+Xb,t[E,S]->F,t[E,S2]->Xb,t[E,V2]->D+F,t[E,V3]->A+B+C,t[E,W]->B,t[E,X]->B,t[E,Z]->A+M,t[F,N]->F,t[F,P]->D+Xb,t[F,Q]->D+F,t[F,S]->Xb,t[F,S2]->E,t[F,V2]->D+E,t[F,V3]->A+B+C,t[F,W]->A+M,t[F,X]->B,t[F,Z]->B,t[M,A]->A,t[M,B]->B,t[M,C]->C,t[M,D]->D,t[M,E]->E,t[M,F]->F,t[M,M]->M,t[M,Xb]->Xb,t[N,N]->N,t[N,P]->P,t[N,Q]->Q,t[N,S]->S,t[N,S2]->S2,t[N,V2]->V2,t[N,V3]->V3,t[N,W]->W,t[N,X]->X,t[N,Z]->Z,t[P,N]->P,t[P,P]->N+P+Q+V2,t[P,Q]->P+Q+S2+V2,t[P,S]->V2,t[P,S2]->Q,t[P,V2]->P+Q+S+V2,t[P,V3]->2 V3+W+X+Z,t[P,W]->V3+X,t[P,X]->V3+W,t[P,Z]->V3+Z,t[Q,N]->Q,t[Q,P]->P+Q+S+V2,t[Q,Q]->N+P+Q+V2,t[Q,S]->P,t[Q,S2]->V2,t[Q,V2]->P+Q+S2+V2,t[Q,V3]->2 V3+W+X+Z,t[Q,W]->V3+W,t[Q,X]->V3+Z,t[Q,Z]->V3+X,t[S,N]->S,t[S,P]->Q,t[S,Q]->V2,t[S,S]->S2,t[S,S2]->N,t[S,V2]->P,t[S,V3]->V3,t[S,W]->Z,t[S,X]->W,t[S,Z]->X,t[S2,N]->S2,t[S2,P]->V2,t[S2,Q]->P,t[S2,S]->N,t[S2,S2]->S,t[S2,V2]->Q,t[S2,V3]->V3,t[S2,W]->X,t[S2,X]->Z,t[S2,Z]->W,t[V2,N]->V2,t[V2,P]->P+Q+S2+V2,t[V2,Q]->P+Q+S+V2,t[V2,S]->Q,t[V2,S2]->P,t[V2,V2]->N+P+Q+V2,t[V2,V3]->2 V3+W+X+Z,t[V2,W]->V3+Z,t[V2,X]->V3+X,t[V2,Z]->V3+W,t[V3,A]->2 V3+W+X+Z,t[V3,B]->3 V3+W+X+Z,t[V3,C]->V3+W+X+Z,t[V3,D]->N+2 P+2 Q+S+S2+2 V2,t[V3,E]->P+Q+V2,t[V3,F]->P+Q+V2,t[V3,M]->V3,t[V3,Xb]->P+Q+V2,t[W,A]->V3+W,t[W,B]->V3+X+Z,t[W,C]->V3,t[W,D]->P+Q+V2,t[W,E]->S2+V2,t[W,F]->N+Q,t[W,M]->W,t[W,Xb]->P+S,t[X,A]->V3+X,t[X,B]->V3+W+Z,t[X,C]->V3,t[X,D]->P+Q+V2,t[X,E]->Q+S,t[X,F]->P+S2,t[X,M]->X,t[X,Xb]->N+V2,t[Xb,N]->Xb,t[Xb,P]->D+F,t[Xb,Q]->D+E,t[Xb,S]->E,t[Xb,S2]->F,t[Xb,V2]->D+Xb,t[Xb,V3]->A+B+C,t[Xb,W]->B,t[Xb,X]->A+M,t[Xb,Z]->B,t[Z,A]->V3+Z,t[Z,B]->V3+W+X,t[Z,C]->V3,t[Z,D]->P+Q+V2,t[Z,E]->N+P,t[Z,F]->S+V2,t[Z,M]->Z,t[Z,Xb]->Q+S2,Lp[V2,V2]->P,Rp[V2,V2]->P,Lp[V2,P]->V2,Rp[V2,P]->P,Lp[V2,Q]->Q,Rp[V2,Q]->P,Lp[V2,S]->S2,Rp[V2,S]->V2,Lp[V2,S2]->S,Rp[V2,S2]->V2,Lp[P,V2]->V2,Rp[P,V2]->Q,Lp[P,P]->Q,Rp[P,P]->Q,Lp[P,Q]->P,Rp[P,Q]->Q,Lp[P,S]->S2,Rp[P,S]->P,Lp[P,S2]->S,Rp[P,S2]->P,Lp[Q,V2]->Q,Rp[Q,V2]->V2,Lp[Q,P]->P,Rp[Q,P]->V2,Lp[Q,Q]->V2,Rp[Q,Q]->V2,Lp[Q,S]->S2,Rp[Q,S]->Q,Lp[Q,S2]->S,Rp[Q,S2]->Q,Lp[S,V2]->Q,Rp[S,V2]->S,Lp[S,P]->V2,Rp[S,P]->S,Lp[S,Q]->P,Rp[S,Q]->S,Lp[S2,V2]->P,Rp[S2,V2]->S2,Lp[S2,P]->Q,Rp[S2,P]->S2,Lp[S2,Q]->V2,Rp[S2,Q]->S2},(_[A_,B_]->C_)/;MemberQ[bimodules[N,N],A]∧MemberQ[bimodules[N,N],B]∧MemberQ[bimodules[N,N],C]]

Simplify[quasiTriangularityEquations//.tensorRules~Join~braidingRules1]

Simplify[quasiTriangularityEquations//.tensorRules~Join~braidingRules2]

Simplify[quasiTriangularityEquations//.tensorRules~Join~braidingRules3]

Simplify[quasiTriangularityEquations//.tensorRules~Join~braidingRules4]

(Outer[List,bimodules[N,N],bimodules[N,N]])/.{A_,B_}:>{{Lp[A,B],Rp[A,B]},{A,B}}/.braidingRules1//TableForm

(Outer[List,bimodules[N,N],bimodules[N,N]])/.{A_,B_}:>{{Lp[A,B],Rp[A,B]},{A,B}}/.braidingRules2//TableForm

(Outer[List,bimodules[N,N],bimodules[N,N]])/.{A_,B_}:>{{Lp[A,B],Rp[A,B]},{A,B}}/.braidingRules3//TableForm

(Outer[List,bimodules[N,N],bimodules[N,N]])/.{A_,B_}:>{{Lp[A,B],Rp[A,B]},{A,B}}/.braidingRules4//TableForm

braidingRules1-braidingRules2

Sort[braidingRules1]-Sort[braidingRules3]

Sort[braidingRules3]-Sort[braidingRules4]

Sort[braidingRules1]-Sort[braidingRules4]

dualityRules={dual[A]->A,dual[B]->B,dual[C]->C,dual[D]->V3,dual[E]->Z,dual[F]->W,dual[M]->M,dual[N]->N,dual[P]->P,dual[Q]->Q,dual[S]->S2,dual[S2]->S,dual[V2]->V2,dual[V3]->D,dual[W]->F,dual[X]->Xb,dual[Xb]->X,dual[Z]->E}

tridentEquation[a_,b_]:={Rp[a,dual[b]]==Lp[dual[Lp[a,dual[b]]],a],b==Rp[dual[Lp[a,dual[b]]],a]}

tridentEquations=tridentEquation@@#&/@Flatten[Outer[List,bimodules[N,N],bimodules[N,N]],1]

tridentEquations//.dualityRules~Join~braidingRules1

tridentEquations//.dualityRules~Join~braidingRules3

tridentEquations//.dualityRules~Join~braidingRules4