Difference between revisions of "ApCoCoA-1:BBSGen.JacobiLin"
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<command> | <command> | ||
<title>BBSGen.JacobiLin</title> | <title>BBSGen.JacobiLin</title> | ||
| − | <short_description>This function computes the K[c]-linear polynomial entries of the Jacobi identity | + | <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> | ||
| − | Let R:=K[x_1,...,x_N] and A_k be the generic multiplication matrix associated to x_k. Let | + | 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}. |
| − | + | <par/> | |
| − | Let m,k,l in {1,...,N}. This function computes the polynomial entries of the Jacobi identity J^ | + | 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. | Please note that this function does not work for the case, where N=2. | ||
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<item>@param Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N]. | <item>@param 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^ | + | <item>@return The K[c]-linear entries of the Jacobi Identity J^ikl. |
.</item> | .</item> | ||
</itemize> | </itemize> | ||
| Line 45: | Line 46: | ||
</description> | </description> | ||
<types> | <types> | ||
| − | <type> | + | <type>bbsmingensys</type> |
| − | <type> | + | <type>poly</type> |
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
</types> | </types> | ||
| − | <see>BBSGen.JacobiFull</see> | + | <see>ApCoCoA-1:BBSGen.JacobiFull|BBSGen.JacobiFull</see> |
| − | <see>BBSGen.JacobiStep</see> | + | <see>ApCoCoA-1:BBSGen.JacobiStep|BBSGen.JacobiStep</see> |
<key>BBSGen.JacobiLin</key> | <key>BBSGen.JacobiLin</key> | ||
| + | <key>JacobiLin</key> | ||
| − | <wiki-category>Package_bbsmingensys</wiki-category> | + | <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]]]]
