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

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(New page: <command> <title>BBSGen.JacobiStep</title> <short_description>: This function computes the polynomial entry of a given position of the Jacobi identity [ A_i[A_k,A_l ] ]+[ A_k[ A_l...)
 
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{{Version|1}}
 
<command>
 
<command>
 
   <title>BBSGen.JacobiStep</title>
 
   <title>BBSGen.JacobiStep</title>
−
   <short_description>: 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}.
+
   <short_description>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 ] ] where m,k,l is from {1,...,N} and I,J in {1,...,Len(OO)}.
 
</short_description>
 
</short_description>
 
    
 
    
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BBGGen.JacobiStep(I,J,OO,BO,N);
 
BBGGen.JacobiStep(I,J,OO,BO,N);
−
BBSGen.JacobiStep(I:INTEGER,J:INTEGER,OO:LIST,BO:LIST,N:INTEGER):POLY
+
BBSGen.JacobiStep(I:INTEGER,J:INTEGER,OO:LIST,BO:LIST,N:INTEGER):LIST
 
</syntax>
 
</syntax>
 
   <description>
 
   <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}.
+
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=K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]], where Mu:=Len(OO),Nu:=Len(BO) and N is the number of indeterminates from the polynomial ring K[x_1,...,x_N].
  
  
  
−
Please note that this function does not work for the case, where n=2.  
+
Please note that this function does not work for the case, where N=2.  
  
  
 
<itemize>
 
<itemize>
−
   <item>@param  I,J position of J^{klm}, order ideal OO, border BO, the number of Indeterminates of the Polynomial.
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   <item>@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].
−
 
 
 
</item>
 
</item>
−
   <item>@return  The K[c]-linear entries of the Jacobi Identity J^{ikl}.
+
   <item>@return  The list of the polynomial in the (I,J) position of the Jacobi Identity J^{ikl}.
 
</item>
 
</item>
 
</itemize>
 
</itemize>
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OO:=[1,x[1]];
 
OO:=[1,x[1]];
−
BO:=BB.Border(OO);
+
BO:=$apcocoa/borderbasis.Border(OO);
 
Mu:=Len(OO);
 
Mu:=Len(OO);
 
Nu:=Len(BO);
 
Nu:=Len(BO);
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BBSGen.JacobiStep(1,2,OO,BO,N);
 
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]]]]
+
[[-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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   </description>
 
   </description>
 
   <types>
 
   <types>
−
     <type>borderbasis</type>
+
     <type>bbsmingensys</type>
−
     <type>ideal</type>
+
     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
−
<see>BB.Border</see>
 
−
  <see>BB.Box</see>
 
−
<see>BBSGen.JacobiFull</see>
 
−
<see>BBSGen.JacobiLin</see>
 
  
−
   <key>Wmat</key>
+
<see>ApCoCoA-1:BBSGen.JacobiFull|BBSGen.JacobiFull</see>
−
   <key>BBSGen.Wmat</key>
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<see>ApCoCoA-1:BBSGen.JacobiLin|BBSGen.JacobiLin</see>
−
   <key>bbsmingensys.Wmat</key>
+
 
−
   <wiki-category>Package_bbsmingensys</wiki-category>
+
   <key>JacobiStep</key>
 +
   <key>BBSGen.JacobiStep</key>
 +
   <key>bbsmingensys.JacobiStep</key>
 +
   <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.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 ] ] 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):LIST

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=K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]], where Mu:=Len(OO),Nu:=Len(BO) and N is the number of indeterminates from the polynomial ring K[x_1,...,x_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 ring K[x_1,...,x_N].

  • @return The list of 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




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