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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
SandBox Sedenion Algebra is Frobenius In Just One Way ←	↑	→ The Algebra of Complex Numbers Is Frobenius In Many Ways

SandBoxFrobeniusAlgebra

last edited 12 years ago by Bill Page

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Editor: Bill Page Time: 2011/02/11 17:47:40 GMT-8
Note: new

changed:
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An n-dimensional algebra is represented by a tensor $Y=\{ y_{ij}^k \} \ i,j,k =1,2, ... n $ viewed as an operator with two inputs 'i,j' and one output 'k'.
\begin{axiom}
n:=2
T:=CartesianTensor(1,n,EXPR INT)
Y:=unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )$T
\end{axiom}
Given two vectors 'U' and 'V'
\begin{axiom}
U:=unravel([script(u,[[],[i]]) for i in 1..n])$T
V:=unravel([script(v,[[],[i]]) for i in 1..n])$T
\end{axiom}
the tensor 'Y' operates on their tensor product
\begin{axiom}
UV:=product(U,V)
YUV:=product(Y,UV)
YUV.[1,1,1,1,2]
YUV.[1,1,1,2,1]
YUV.[1,1,2,1,1]
YUV.[1,2,1,1,1]
YUV.[2,1,1,1,1]
Y*U*V
\end{axiom}

An n-dimensional algebra is represented by a tensor $Y=\{ y_{ij}^k \} \ i,j,k =1,2, ... n$ viewed as an operator with two inputs i,j and one output k.

axiom

n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

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

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

(2)

Type: Domain

axiom

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

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

(3)

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

Given two vectors U and V

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,Expression(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,Expression(Integer))

the tensor Y operates on their tensor product

axiom

UV:=product(U,V)

$\label{eq6}\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}}&{{u^{1}}\ {v^{2}}} \ {{u^{2}}\ {v^{1}}}&{{u^{2}}\ {v^{2}}}$

(6)

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

axiom

YUV:=product(Y,UV)

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}} \ {{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{2}}\ {v^{2}}\ {y_{1, \: 1}^{1}}}$

(7)

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

axiom

YUV.[1,1,1,1,2]

$\label{eq8}{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}$

(8)

Type: Expression(Integer)

axiom

YUV.[1,1,1,2,1]

$\label{eq9}{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}$

(9)

Type: Expression(Integer)

axiom

YUV.[1,1,2,1,1]

$\label{eq10}{u^{1}}\ {v^{1}}\ {y_{2, \: 1}^{1}}$

(10)

Type: Expression(Integer)

axiom

YUV.[1,2,1,1,1]

$\label{eq11}{u^{1}}\ {v^{1}}\ {y_{1, \: 2}^{1}}$

(11)

Type: Expression(Integer)

axiom

YUV.[2,1,1,1,1]

$\label{eq12}{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}$

(12)

Type: Expression(Integer)

axiom

Y*U*V

$\label{eq13}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(13)

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