Difference between revisions of "ApCoCoA-1:BBSGen.JacobiStep"

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<see>BBSGen.JacobiLin</see>
 
<see>BBSGen.JacobiLin</see>
  
   <key>Wmat</key>
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   <key>JacobiStep</key>
   <key>BBSGen.Wmat</key>
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   <key>BBSGen.JacobiStep</key>
   <key>bbsmingensys.Wmat</key>
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   <key>bbsmingensys.JacobiStep</key>
 
   <wiki-category>Package_bbsmingensys</wiki-category>
 
   <wiki-category>Package_bbsmingensys</wiki-category>
 
</command>
 
</command>

Revision as of 18:29, 18 June 2012

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 ] ]=0 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):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 ring K[x_1,...,x_N].

  • @return 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]]]]
-----------


BBSGen.JacobiFull

BBSGen.JacobiLin