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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
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SandBoxFrobeniusAlgebra

last edited 13 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

axiom

)library DEXPR

   DistributedExpression is now explicitly exposed in frame initial
   DistributedExpression will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

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

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

(2)

Type: Domain

axiom

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

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1}^{1, \: 1}}&{y_{1}^{1, \: 2}} \ {y_{1}^{2, \: 1}}&{y_{1}^{2, \: 2}}$

(3)

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

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

axiom

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

$\label{eq4}\left[{u_{1}}, \:{u_{2}}\right]$

(4)

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

axiom

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

$\label{eq5}\left[{v_{1}}, \:{v_{2}}\right]$

(5)

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

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

axiom

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

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{{{y_{1}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{1}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{1}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{2}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{2}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}\right]$

(6)

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

or in a more convenient notation:

axiom

W:=(Y*U)*V

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{y_{1}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{1}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{1}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{2}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{2}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}\right]$

(7)

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

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

axiom

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

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

(8)

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

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

axiom

C:=groebner(ravel(K))

$\label{eq9}\left[{{y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}}, \:{{y_{1}^{2, \: 1}}-{y_{1}^{1, \: 2}}}\right]$

(9)

Type: List(Polynomial(Integer))

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.

axiom

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

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \:{-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}^2}+{{y_{1}^{2, \: 1}}^2}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{1}^{2, \: 2}}}+{{y_{1}^{2, \: 1}}\ {y_{1}^{2, \: 2}}}}, \:{{{y_{2}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}+{{y_{2}^{1, \: 2}}^2}-{{y_{2}^{2, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \:{{{y_{2}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{2}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{y_{2}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}\right]$

(10)

Type: List(DistributedExpression?(Integer))