SetDirectory["~/kappaLib/"];
<<kappaLib-1.2.m
<<helper.m

Verify the identities used to prove Theorem 4.5

We assume that A,B are arbitrary antisymmetric (2,0)-tensors. Say

    A = 1/2 A^ij d/dx^i /\ d/dx^j
    
and coefficients A^ij are given by matrix A. Similarly for B.

A=emMatrix["A",4,Structure->"AntiSymmetric"];
B=emMatrix["B",4,Structure->"AntiSymmetric"];
A//MatrixForm
B//MatrixForm

(* kappa = general (2,2)-tensor *)
kappa=emGeneralKappa["k"];

(* contract kappa with a bivector from the left *)
contract[biv_,kappa_]:=Table[
1/2 Sum[
biv[[i]][[j]]emReadNormal[kappa,a,b,i,j]
,
{i,1,4},{j,1,4}
]
,
{a,1,4},{b,1,4}
]

Identities

delta=emPoincare[emBiProduct[density,A,B]]-emBiProduct[density,B,A];
Union[Flatten[Simplify[delta]]]

LHS=emCompose[emBiProduct[density,A,B],kappa];RHS=emBiProduct[density,A,contract[B,kappa]];
Union[Flatten[Simplify[RHS-LHS]]]

LHS=emCompose[emBiProduct[density,A,B],kappa];RHS=emBiProduct[density,A,contract[B,kappa]];
Union[Flatten[Simplify[RHS-LHS]]]

LHS=emCompose[kappa,emBiProduct[density,A,B]];RHS=emBiProduct[density,contract[A,emPoincare[kappa]],B];
Union[Flatten[Simplify[RHS-LHS]]]

LHS=emCompose[emBiProduct[density,A,B],emBiProduct[density,B,A]];RHS=emTrace[emBiProduct[density,B,B]]emBiProduct[density,A,A];
Union[Flatten[Simplify[RHS-LHS]]]

LHS=emPoincare[emPoincare[kappa]];
RHS=kappa;
Union[Flatten[Simplify[RHS-LHS]]]

Local expression for density bar(B) \otimes B.

FullSimplify[emTrace[emBiProduct[density,B,B]]]