ApCoCoA-1:BBSGen.JacobiLin

BBSGen.JacobiLin

This function computes the K[c]-linear polynomial entries of the Jacobi identity matrix [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ] where m,k,l is from {1,...,N}.

Syntax

```BBSGen.JacobiLin(OO,BO,N);
BBSGen.JacobiLin(OO:LIST,BO:LIST,N:INTEGER):MATRIX
```

Description

Let R:=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let OO be the order ideal and BO its order. Let Mu:=Len(OO) and Nu:= Len(BO). Let tau^kl_ij be the polynomial in the (i,j) position of the [A_k,A_l] where k,l in {1,..,N}. Let m,k,l in {1,...,N}.

This function computes the polynomial entries of the Jacobi identity J^mkl= [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k]] that has constant coeffiecients. During the computation entries of the commutators Tau^kl_ij will be considered as indeterminates t[k,l,i,j] in K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].

Please note that this function does not work for the case, where N=2.

• @param Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].

• @return The K[c]-linear entries of the Jacobi Identity J^ikl. .

Example

```Use R::=QQ[x[1..3]];

OO:=[1,x[1]];
BO:=\$apcocoa/borderbasis.Border(OO);
Mu:=Len(OO);
Nu:=Len(BO);
N:=Len(Indets());
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];

BBSGen.JacobiLin(OO,BO,N);

[[   [    -t[2,3,1,2],0],
[ t[2,3,1,1] - t[2,3,2,2], t[2,3,1,2]]]]

```