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

From ApCoCoAWiki
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{{Version|1}}
 
<command>
 
<command>
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   <title>BBSGen.TraceSyzStep</title>
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   <title>BBSGen.JacobiLin</title>
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   <short_description>:  This function computes  the K[c]-linear polynomial  entries of the Jacobi identity   [ 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}.
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   <short_description>This function computes  the K[c]-linear polynomial  entries of the Jacobi identity matrix [ 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}.
 
</short_description>
 
</short_description>
 
    
 
    
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</syntax>
 
</syntax>
 
   <description>
 
   <description>
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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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Let R:=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let OO be the order ideal and BO its order. Let Mu:=Len(OO) and Nu:= Len(BO). Let  tau^kl_ij be the polynomial in the (i,j) position of the  [A_k,A_l] where k,l in {1,..,N}. Let m,k,l in {1,...,N}.  
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<par/>
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Let m,k,l \in {1,...,n}. This function computes the polynomial 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 ] ] that has constant coeffiecients. During the computation  entries of the commutators Tau^kl_ij will be  considered as indeterminates  t[k,l,i,j]\in XX. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].
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This function computes the polynomial 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]] that has constant coeffiecients. 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]]. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].
  
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Please note that this function does not work for the case, where n=2.  
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Please note that this function does not work for the case, where N=2.  
  
  
 
<itemize>
 
<itemize>
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   <item>@param  Order ideal OO, border BO, the number of Indeterminates of the Polynomial.(see <commandref>BB.Border</commandref> in package borderbasis)
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   <item>@param  Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N].
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</item>
 
</item>
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   <item>@return  The K[c]-linear entries of the Jacobi Identity J^{ikl}.
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   <item>@return  The K[c]-linear entries of the Jacobi Identity J^ikl.
 
.</item>
 
.</item>
 
</itemize>
 
</itemize>
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OO:=[1,x[1]];
 
OO:=[1,x[1]];
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BO:=BB.Border(OO);
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BO:=$apcocoa/borderbasis.Border(OO);
 
Mu:=Len(OO);
 
Mu:=Len(OO);
 
Nu:=Len(BO);
 
Nu:=Len(BO);
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Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
 
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
  
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BBSGen.BoxJacobiLin(OO,BO,N);
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BBSGen.JacobiLin(OO,BO,N);
  
  
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   </description>
 
   </description>
 
   <types>
 
   <types>
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     <type>borderbasis</type>
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     <type>bbsmingensys</type>
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     <type>ideal</type>
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     <type>poly</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
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<see>BB.Border</see>
 
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  <see>BB.Box</see>
 
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<see>BBSGen.JacobiFull</see>
 
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<see>BBSGen.JacobiStep</see>
 
  
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  <key>Wmat</key>
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<see>ApCoCoA-1:BBSGen.JacobiFull|BBSGen.JacobiFull</see>
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   <key>BBSGen.Wmat</key>
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<see>ApCoCoA-1:BBSGen.JacobiStep|BBSGen.JacobiStep</see>
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   <key>bbsmingensys.Wmat</key>
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   <wiki-category>Package_bbsmingensys</wiki-category>
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   <key>BBSGen.JacobiLin</key>
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   <key>JacobiLin</key>
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 +
 
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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.JacobiLin

This function computes the K[c]-linear polynomial entries of the Jacobi identity matrix [ 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.JacobiLin(OO,BO,N);
BBSGen.JacobiLin(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 OO be the order ideal and BO its order. Let Mu:=Len(OO) and Nu:= Len(BO). Let tau^kl_ij be the polynomial in the (i,j) position of the [A_k,A_l] where k,l in {1,..,N}. Let m,k,l in {1,...,N}.

This function computes the polynomial 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]] that has constant coeffiecients. 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]]. Therefore the result of BBSGen.JacobiLin is a list of polynomials from the ring K[t[1..N,1..N,1..Mu,1..Mu]].

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 K[c]-linear 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.JacobiLin(OO,BO,N);


[[   [    -t[2,3,1,2],0],
    [ t[2,3,1,1] - t[2,3,2,2], t[2,3,1,2]]]]


BBSGen.JacobiFull

BBSGen.JacobiStep




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