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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<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}.
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   <short_description>:  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}.  
 
</short_description>
 
</short_description>
 
    
 
    
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</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}.
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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}.  
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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.
  
  
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<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.(see <commandref>BB.Border</commandref> in the package borderbasis)
  
 
</item>
 
</item>
   <item>@return  The K[c]-linear entries of the Jacobi Identity J^{ikl}.
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   <item>@return  The polynomial in the (I,J) of the Jacobi Identity J^{ikl}.
 
</item>
 
</item>
 
</itemize>
 
</itemize>
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     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
<see>BB.Border</see>
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  <see>BB.Box</see>
 
 
<see>BBSGen.JacobiFull</see>
 
<see>BBSGen.JacobiFull</see>
 
<see>BBSGen.JacobiLin</see>
 
<see>BBSGen.JacobiLin</see>

Revision as of 09:25, 8 June 2012

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]]]]
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BBSGen.JacobiFull

BBSGen.JacobiLin