ApCoCoA-1:BBSGen.JacobiStep

BBSGen.JacobiStep

This function computes the polynomial entry of a given position of the Jacobi identity [ A_i[A_k,A_l ] ]+[ A_k[ A_l,A_i]] +[ A_l[A_i,A_k ] ]=0 , where i,k,l is from {1...n}.

Syntax

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

Description

Let R=K[x_1,...,x_n] and A_i is the generic multiplication matrix for x_i. Let Tau^kl_ij :=t[k,l,i,j] be the (i,j) ^th entry of matrix the operation [A_k,A_l]. This function computes the plynomial entry of a given position of the Jacobi identity [ A_m[A_k,A_l ] ]+[ A_k[ A_l,A_m]] +[ A_l[A_m,A_k ] ]=0 , where m,k,l is from {1...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.

  
  

• @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.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]]]]
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BB.Border

BB.Box

BBSGen.JacobiFull

BBSGen.JacobiLin