ApCoCoA-1:BBSGen.JacobiLin

BBSGen.TraceSyzStep

This function computes the K[c]-linear polynomial entries of the Jacobi identity [ A_i[A_k,A_l ] ]+[ A_k[ A_l,A_i]] +[ A_l[A_i,A_k ] ] where i,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 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 XX. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring CoeffRing[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.(see <commandref>BB.Border</commandref> in package borderbasis)

  
  

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

Example

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

OO:=[1,x[1]];
BO:=BB.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.BoxJacobiLin(OO,BO,N);


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

BB.Border

BB.Box

BBSGen.JacobiFull

BBSGen.JacobiStep