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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 13 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 3 of 26

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Editor: Bill Page Time: 2011/02/11 20:07:45 GMT-8
Note: reindex

changed:
-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'.
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'.

added:
reindex(Yijk,[3,1,2])

added:
reindex(Yijk,[3,1,2])*Ui*Vj

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

Yijk:=unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for k in 1..n]
      for j in 1..n]
        for i 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,Expression(Integer))

axiom

reindex(Yijk,[3,1,2])

$\label{eq4}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1, \: 1}^{1}}&{y_{1, \: 2}^{1}} \ {y_{2, \: 1}^{1}}&{y_{2, \: 2}^{1}}$

(4)

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

axiom

Y.[1,1,2]

$\label{eq5}Y_{1, \: 1, \: 2}$

(5)

Type: Symbol

axiom

Y.[1,2,1]

$\label{eq6}Y_{1, \: 2, \: 1}$

(6)

Type: Symbol

axiom

Y.[2,1,1]

$\label{eq7}Y_{2, \: 1, \: 1}$

(7)

Type: Symbol

Given two vectors U and V

axiom

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

$\label{eq8}\left[{u^{1}}, \:{u^{2}}\right]$

(8)

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

axiom

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

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

(9)

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

the tensor Y operates on their tensor product

axiom

UVij:=product(Ui,Vj)

$\label{eq10}\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}}&{{u^{1}}\ {v^{2}}} \ {{u^{2}}\ {v^{1}}}&{{u^{2}}\ {v^{2}}}$

(10)

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

axiom

UVij.[1,2]

$\label{eq11}{u^{1}}\ {v^{2}}$

(11)

Type: Expression(Integer)

axiom

UVij.[2,1]

$\label{eq12}{u^{2}}\ {v^{1}}$

(12)

Type: Expression(Integer)

axiom

YUV:=product(Yijk,UVij)

$\label{eq13}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}} \ {{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{2}}\ {v^{2}}\ {y_{1, \: 1}^{1}}}$

(13)

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

axiom

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

$\label{eq14}{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}$

(14)

Type: Expression(Integer)

axiom

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

$\label{eq15}{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}$

(15)

Type: Expression(Integer)

axiom

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

$\label{eq16}{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}$

(16)

Type: Expression(Integer)

axiom

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

$\label{eq17}{u^{1}}\ {v^{1}}\ {y_{1, \: 2}^{1}}$

(17)

Type: Expression(Integer)

axiom

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

$\label{eq18}{u^{1}}\ {v^{1}}\ {y_{2, \: 1}^{1}}$

(18)

Type: Expression(Integer)

axiom

contract(contract(YUV,1,4),1,3)

$\label{eq19}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(19)

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

axiom

contract(contract(Yijk,1,UVij,1),1,3)

$\label{eq20}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(20)

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

axiom

reindex(Yijk,[3,1,2])*Ui*Vj

$\label{eq21}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{2, \: 1}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{2, \: 1}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(21)

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