# ApCoCoA-1:BBSGen.JacobiStep

## BBSGen.JacobiStep

Let R:=K[x_1,...,x_N] and let OO be an order ideal. This function computes the entry in the position (I,J) of the Jacobi identity matrix J^klm [ 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} and I,J in {1,...,Len(OO)}.

### Syntax

```BBGGen.JacobiStep(I,J,OO,BO,N);
BBSGen.JacobiStep(I:INTEGER,J:INTEGER,OO:LIST,BO:LIST,N:INTEGER):LIST
```

### Description

Let R=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let tau^kl_ij be the polynomial in the (i,j) position of the [A_k,A_l] where k,l in {1,..,N}.

This function computes the given (I,J) position of the Jacobi identity J^mkl= [ 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}. During the computation entries of the commutators tau^kl_ij will be considered as indeterminates t[k,l,i,j] in XX=K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]], where Mu:=Len(OO),Nu:=Len(BO) and N is the number of indeterminates from the polynomial ring K[x_1,...,x_N].

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

• @param I,J position of J^klm, order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].

• @return The list of the polynomial in the (I,J) position 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.JacobiStep(1,2,OO,BO,N);

[[-c[1,3]t[1,2,1,1] + c[1,1]t[1,2,1,2] - c[2,3]t[1,2,1,2] +
c[1,3]t[1,2,2,2] - c[1,4]t[1,3,1,1] + c[1,2]t[1,3,1,2] -
c[2,4]t[1,3,1,2] + c[1,4]t[1,3,2,2] - c[1,5]t[2,3,1,1] -
c[2,5]t[2,3,1,2] + c[1,5]t[2,3,2,2]]]]
-----------

```