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Edit detail for SandBoxFrobeniusAlgebra revision 6 of 26

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Editor: Bill Page
Time: 2011/02/11 21:15:23 GMT-8
Note: take 2

changed:
-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 one output 'k'.
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 one output 'k'.

changed:
-  [[[script(y,[[i,j],[k]])
  [[[script(y,[[k],[j,i]])

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

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 to yield a vector W

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
Wk:=(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 one output k.

axiom
Ykji:=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{eq22}\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}}
(22)
Type: CartesianTensor?(1,2,Expression(Integer))

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

\label{eq23}\left[{u_{1}}, \:{u_{2}}\right](23)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
Vj:=unravel([script(v,[[i]]) for i in 1..n])$T

\label{eq24}\left[{v_{1}}, \:{v_{2}}\right](24)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
contract(contract(Ykji,3,product(Ui,Vj),1),2,3)

\label{eq25}\begin{array}{@{}l}
\displaystyle
\left[{{{\left({{y_{1}^{2, \: 2}}\ {u_{2}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{1}^{1, \: 2}}\ {u_{2}}}+{{y_{1}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}, \: \right.
\
\
\displaystyle
\left.{{{\left({{y_{2}^{2, \: 2}}\ {u_{2}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{2}^{1, \: 2}}\ {u_{2}}}+{{y_{2}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}\right] 
(25)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
Wk:=(Ykji*Ui)*Vj

\label{eq26}\begin{array}{@{}l}
\displaystyle
\left[{{{\left({{y_{1}^{2, \: 2}}\ {u_{2}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{1}^{1, \: 2}}\ {u_{2}}}+{{y_{1}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}, \: \right.
\
\
\displaystyle
\left.{{{\left({{y_{2}^{2, \: 2}}\ {u_{2}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}}\right)}\ {v_{2}}}+{{\left({{y_{2}^{1, \: 2}}\ {u_{2}}}+{{y_{2}^{1, \: 1}}\ {u_{1}}}\right)}\ {v_{1}}}}\right] 
(26)
Type: CartesianTensor?(1,2,Expression(Integer))