ApCoCoA-1:BBSGen.JacobiFull
BBSGen.TraceSyzStep
Let R:=K[x_1,...,x_N]. This function computes the entries 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 ] ]=0 , where m,k,l is from {1...N}.
Syntax
BBSGen.JacobiFull(OO,BO,N); BBSGen.JacobiFull(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}.
This function computes the 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 ] ] , 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.
When the polynomial entries of the above matrix are large, one may not have a result. In that case we recommend JacobiStep or JacobiLin.
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 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.JacobiFull(OO,BO,N); [[ [c[1,1]t[1,2,1,1] + c[1,3]t[1,2,2,1] + c[1,2]t[1,3,1,1] + c[1,4]t[1,3,2,1] + c[1,5]t[2,3,2,1], c[1,1]t[1,2,1,2] + c[1,3]t[1,2,2,2] + c[1,2]t[1,3,1,2] + c[1,4]t[1,3,2,2] + c[1,5]t[2,3,2,2]], [ c[2,1]t[1,2,1,1] + c[2,3]t[1,2,2,1] + c[2,2]t[1,3,1,1] + c[2,4]t[1,3,2,1] + c[2,5]t[2,3,2,1] + t[2,3,1,1], c[2,1]t[1,2,1,2] + c[2,3]t[1,2,2,2] + c[2,2]t[1,3,1,2] + c[2,4]t[1,3,2,2] + c[2,5]t[2,3,2,2] + t[2,3,1,2]]]]