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

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<command>
 
   <title>BBSGen.JacobiFull</title>
 
   <title>BBSGen.JacobiFull</title>
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</syntax>
 
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   <description>
 
   <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}.  
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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 K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]].
 
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 K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]].
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   </description>
 
   </description>
 
   <types>
 
   <types>
     <type>borderbasis</type>
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     <type>bbsmingensys</type>
     <type>ideal</type>
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     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
  
<see>BBSGen.JacobiStep</see>
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<see>ApCoCoA-1:BBSGen.JacobiStep|BBSGen.JacobiStep</see>
<see>BBSGen.JacobiLin</see>
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<see>ApCoCoA-1:BBSGen.JacobiLin|BBSGen.JacobiLin</see>
 
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<key>JacobiFull</key>
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  <key>BBSGen.JacobiFull</key>
 
    
 
    
   <wiki-category>Package_bbsmingensys</wiki-category>
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   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:50, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.JacobiFull

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 ] ], 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 K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]].

 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]]]]


BBSGen.JacobiStep

BBSGen.JacobiLin