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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 4 of 26

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Editor: Bill Page Time: 2011/02/11 20:25:29 GMT-8
Note: reindex

changed:
-Y.[1,1,2]
-Y.[1,2,1]
-Y.[2,1,1]
Yijk[1,1,2]
Yijk[1,2,1]
Yijk[2,1,1]

changed:
-UVij.[1,2]
-UVij.[2,1]
UVij[1,2]
UVij[2,1]

changed:
-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]
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]

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

added:

Take 2

An n-dimensional algebra is represented by a tensor $Y=\{ {y^k}_{ji} \} \ i,j,k =1,2, ... n $ viewed as an operator with two inputs 'i,j' and output 'k'.
\begin{axiom}
n:=2
T:=CartesianTensor(1,n,EXPR INT)
Ykji:=unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )$T
Ykji[1,1,2]
Ykji[1,2,1]
Ykji[2,1,1]
\end{axiom}
Given two vectors 'U' and 'V'
\begin{axiom}
Ui:=unravel([script(u,[[],[i]]) for i in 1..n])$T
Vj:=unravel([script(v,[[],[i]]) for i in 1..n])$T
\end{axiom}
the tensor 'Y' operates on their tensor product
\begin{axiom}
UVij:=product(Ui,Vj)
UVij[1,2]
UVij[2,1]
YUV:=product(Ykji,UVij)
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]
contract(contract(YUV,3,4),2,3)
contract(contract(Ykji,3,UVij,1),2,3)
Ykji*Ui*Vj
\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

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

Yijk[1,1,2]

$\label{eq5}y_{1, \: 1}^{2}$

(5)

Type: Expression(Integer)

axiom

Yijk[1,2,1]

$\label{eq6}y_{1, \: 2}^{1}$

(6)

Type: Expression(Integer)

axiom

Yijk[2,1,1]

$\label{eq7}y_{2, \: 1}^{1}$

(7)

Type: Expression(Integer)

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,[3,2,1])*Ui*Vj

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

(21)

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

Take 2

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

axiom

n:=2

$\label{eq22}2$

(22)

Type: PositiveInteger?

axiom

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

$\label{eq23}\hbox{\axiomType{CartesianTensor}\ } (1, 2, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(23)

Type: Domain

axiom

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

$\label{eq24}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1, \: 1}^{1}}&{y_{2, \: 1}^{1}} \ {y_{1, \: 2}^{1}}&{y_{2, \: 2}^{1}}$

(24)

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

axiom

Ykji[1,1,2]

$\label{eq25}y_{2, \: 1}^{1}$

(25)

Type: Expression(Integer)

axiom

Ykji[1,2,1]

$\label{eq26}y_{1, \: 2}^{1}$

(26)

Type: Expression(Integer)

axiom

Ykji[2,1,1]

$\label{eq27}y_{1, \: 1}^{2}$

(27)

Type: Expression(Integer)

Given two vectors U and V

axiom

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

$\label{eq28}\left[{u^{1}}, \:{u^{2}}\right]$

(28)

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

axiom

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

$\label{eq29}\left[{v^{1}}, \:{v^{2}}\right]$

(29)

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

the tensor Y operates on their tensor product

axiom

UVij:=product(Ui,Vj)

$\label{eq30}\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}}&{{u^{1}}\ {v^{2}}} \ {{u^{2}}\ {v^{1}}}&{{u^{2}}\ {v^{2}}}$

(30)

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

axiom

UVij[1,2]

$\label{eq31}{u^{1}}\ {v^{2}}$

(31)

Type: Expression(Integer)

axiom

UVij[2,1]

$\label{eq32}{u^{2}}\ {v^{1}}$

(32)

Type: Expression(Integer)

axiom

YUV:=product(Ykji,UVij)

$\label{eq33}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}} \ {{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{2}}\ {v^{2}}\ {y_{1, \: 1}^{1}}}$

(33)

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

axiom

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

$\label{eq34}{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}$

(34)

Type: Expression(Integer)

axiom

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

$\label{eq35}{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}$

(35)

Type: Expression(Integer)

axiom

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

$\label{eq36}{u^{1}}\ {v^{1}}\ {y_{2, \: 1}^{1}}$

(36)

Type: Expression(Integer)

axiom

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

$\label{eq37}{u^{1}}\ {v^{1}}\ {y_{1, \: 2}^{1}}$

(37)

Type: Expression(Integer)

axiom

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

$\label{eq38}{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}$

(38)

Type: Expression(Integer)

axiom

contract(contract(YUV,3,4),2,3)

$\label{eq39}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(39)

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

axiom

contract(contract(Ykji,3,UVij,1),2,3)

$\label{eq40}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(40)

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

axiom

Ykji*Ui*Vj

$\label{eq41}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(41)

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