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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
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SandBoxFrobeniusAlgebra

last edited 15 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 8 of 26

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Editor: Bill Page Time: 2011/02/12 22:23:47 GMT-8
Note: commutator and associator

added:
The algebra 'Y' is commutative if the following tensor
(the commutator) is zero
\begin{axiom}
K:=Y-reindex(Y,[1,3,2])
\end{axiom}
A basis for the ideal defined by the coefficients of the
commutator is given by:
\begin{axiom}
C:=groebner(ravel(K))
\end{axiom}
An algebra is associative if::

  Y I  =  I Y
   Y       Y

  Note: right figure is mirror image of left!

  1  2 5   1 4  5
   \/ /     \ \/ 
    \/   =   \/  
     \       /   
      6     3    

In other words an algebra is associative if and only
if the following (3,1)-tensor
$A=\{ {a_s}^{kji} =  {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \}$
is zero.
\begin{axiom}
A := Y*Y - reindex(Y,[1,3,2])*reindex(Y,[1,3,2]); ravel(A)
\end{axiom}

An n-dimensional algebra is represented by a (1,2)-tensor $Y=\{ {y_k}^{ji} \} \ i,j,k =1,2, ... n$ viewed as an operator with two inputs i,j and one output k. For example in 2 dimensions

fricas

(1) -> )library DEXPR

   DistributedExpression is now explicitly exposed in frame initial 
   DistributedExpression will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

fricas

T:=CartesianTensor(1,n,DEXPR INT)

$\label{eq2}\hbox{\axiomType{CartesianTensor}\ } \left({1, \: 2, \:{\hbox{\axiomType{DistributedExpression}\ } \left({\hbox{\axiomType{Integer}\ }}\right)}}\right)$

(2)

Type: Type

fricas

Y:=unravel(concat concat
  [[[script(y,[[k],[j,i]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )$T

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {x_{1}}&{x_{1}} \ {x_{1}}&{x_{1}}$

(3)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

Given two vectors $U=\{ u_i \}$ and $V=\{ v_j \}$

fricas

U:=unravel([script(u,[[i]]) for i in 1..n])$T

$\label{eq4}\left[{x_{1}}, \:{x_{1}}\right]$

(4)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

fricas

V:=unravel([script(v,[[i]]) for i in 1..n])$T

$\label{eq5}\left[{x_{1}}, \:{x_{1}}\right]$

(5)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

the tensor Y operates on their tensor product to yield a vector $W=\{ w_k = {y_k}^{ji} u_i v_j \}$

fricas

W:=contract(contract(Y,3,product(U,V),1),2,3)

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{{{x_{4}}\ {x_{6}}\ {x_{8}}}+{{x_{2}}\ {x_{5}}\ {x_{8}}}+{{x_{3}}\ {x_{6}}\ {x_{7}}}+{{x_{1}}\ {x_{5}}\ {x_{7}}}}, \: \right. \ \ \displaystyle \left.{{{x_{4}}\ {x_{6}}\ {x_{8}}}+{{x_{2}}\ {x_{5}}\ {x_{8}}}+{{x_{3}}\ {x_{6}}\ {x_{7}}}+{{x_{1}}\ {x_{5}}\ {x_{7}}}}\right]$

(6)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

or in a more convenient notation:

fricas

W:=(Y*U)*V

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{x_{4}}\ {x_{6}}\ {x_{8}}}+{{x_{2}}\ {x_{5}}\ {x_{8}}}+{{x_{3}}\ {x_{6}}\ {x_{7}}}+{{x_{1}}\ {x_{5}}\ {x_{7}}}}, \: \right. \ \ \displaystyle \left.{{{x_{4}}\ {x_{6}}\ {x_{8}}}+{{x_{2}}\ {x_{5}}\ {x_{8}}}+{{x_{3}}\ {x_{6}}\ {x_{7}}}+{{x_{1}}\ {x_{5}}\ {x_{7}}}}\right]$

(7)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

The algebra Y is commutative if the following tensor (the commutator) is zero

fricas

K:=Y-reindex(Y,[1,3,2])

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{{x_{2}}-{x_{1}}} \ {-{x_{2}}+{x_{1}}}& 0$

(8)

Type: CartesianTensor?(1,2,DistributedExpression?(Integer))

A basis for the ideal defined by the coefficients of the commutator is given by:

fricas

C:=groebner(ravel(K))

   There are 2 exposed and 6 unexposed library operations named 
      groebner having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                            )display op groebner
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      groebner with argument type(s) 
                    List(DistributedExpression(Integer))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

An algebra is associative if:

  Y I  =  I Y
   Y       Y

  Note: right figure is mirror image of left!

  1  2 5   1 4  5
   \/ /     \ \/ 
    \/   =   \/  
     \       /   
      6     3

In other words an algebra is associative if and only if the following (3,1)-tensor $A=\{ {a_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_s}^{ri} {y_r}^{kj} \}$ is zero.

fricas

A := Y*Y - reindex(Y,[1,3,2])*reindex(Y,[1,3,2]); ravel(A)

$\label{eq9}\begin{array}{@{}l} \displaystyle \left[{{{x_{1}}\ {x_{3}}}-{{x_{1}}\ {x_{2}}}}, \:{{{x_{4}}\ {x_{5}}}-{{x_{3}}\ {x_{5}}}+{{x_{2}}\ {x_{4}}}-{{x_{1}}\ {x_{3}}}}, \: \right. \ \ \displaystyle \left.{-{{x_{4}}\ {x_{5}}}+{{x_{3}}\ {x_{5}}}+{{x_{1}}\ {x_{4}}}-{{x_{2}}\ {x_{3}}}}, \:{{{x_{1}}\ {x_{3}}}-{{x_{1}}\ {x_{2}}}}, \: \right. \ \ \displaystyle \left.{-{{x_{2}}\ {x_{3}}}+{{x_{1}}\ {x_{3}}}}, \:{{{x_{2}}\ {x_{3}}}-{{x_{1}}\ {x_{3}}}}, \: \right. \ \ \displaystyle \left.{-{{x_{5}}^{2}}+{{x_{4}}^{2}}-{{x_{2}}\ {x_{3}}}+{{x_{1}}\ {x_{3}}}}, \:{-{{x_{1}}\ {x_{3}}}+{{x_{1}}\ {x_{2}}}}, \: \right. \ \ \displaystyle \left.{{{x_{2}}\ {x_{3}}}-{{x_{1}}\ {x_{3}}}}, \:{{{x_{3}}\ {x_{5}}}-{{x_{3}}\ {x_{4}}}+{{x_{2}}^{2}}-{{x_{1}}^{2}}}, \: \right. \ \ \displaystyle \left.{-{{x_{1}}\ {x_{3}}}+{{x_{1}}\ {x_{2}}}}, \:{{{x_{1}}\ {x_{3}}}-{{x_{1}}\ {x_{2}}}}, \:{-{{x_{2}}\ {x_{3}}}+{{x_{1}}\ {x_{3}}}}, \: \right. \ \ \displaystyle \left.{{{x_{2}}\ {x_{5}}}-{{x_{3}}\ {x_{4}}}+{{x_{1}}\ {x_{3}}}-{{x_{1}}\ {x_{2}}}}, \: \right. \ \ \displaystyle \left.{-{{x_{3}}\ {x_{5}}}+{{x_{2}}\ {x_{4}}}-{{x_{1}}\ {x_{3}}}+{{x_{1}}\ {x_{2}}}}, \:{-{{x_{2}}\ {x_{3}}}+{{x_{1}}\ {x_{3}}}}\right]$

(9)

Type: List(DistributedExpression?(Integer))