ApCoCoA-1:BBSGen.JacobiStep
BBSGen.JacobiStep
- 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 ] ]=0 where m,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_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.
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.(see <commandref>BB.Border</commandref> in the package borderbasis)
@return The polynomial in the (I,J) 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]]]] -----------
