MathAction SandBoxFrobeniusAlgebra

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Editor: Bill Page Time: 2011/02/11 19:58:11 GMT-8
Note: index ordering

changed:
-Y:=unravel(concat concat
Yijk:=unravel(concat concat

changed:
-    for i in 1..n]
    for k in 1..n]

changed:
-        for k in 1..n]
        for i in 1..n]

added:
Y.[1,1,2]
Y.[1,2,1]
Y.[2,1,1]

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

changed:
-UV:=product(U,V)
-YUV:=product(Y,UV)
UVij:=product(Ui,Vj)
UVij.[1,2]
UVij.[2,1]
YUV:=product(Yijk,UVij)

changed:
-Y*U*V
contract(contract(YUV,1,4),1,3)
contract(contract(Yijk,1,UVij,1),1,3)

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

Y.[1,1,2]

$\label{eq4}Y_{1, \: 1, \: 2}$

(4)

Type: Symbol

axiom

Y.[1,2,1]

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

(5)

Type: Symbol

axiom

Y.[2,1,1]

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

(6)

Type: Symbol

Given two vectors U and V

axiom

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

$\label{eq7}\left[{u^{1}}, \:{u^{2}}\right]$

(7)

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

axiom

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

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

(8)

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

the tensor Y operates on their tensor product

axiom

UVij:=product(Ui,Vj)

$\label{eq9}\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}}&{{u^{1}}\ {v^{2}}} \ {{u^{2}}\ {v^{1}}}&{{u^{2}}\ {v^{2}}}$

(9)

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

axiom

UVij.[1,2]

$\label{eq10}{u^{1}}\ {v^{2}}$

(10)

Type: Expression(Integer)

axiom

UVij.[2,1]

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

(11)

Type: Expression(Integer)

axiom

YUV:=product(Yijk,UVij)

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {\left[ \begin{array}{cc} {{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}} \ {{u^{2}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}&{{u^{2}}\ {v^{2}}\ {y_{1, \: 1}^{1}}}$

(12)

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

axiom

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

$\label{eq13}{u^{1}}\ {v^{2}}\ {y_{1, \: 1}^{1}}$

(13)

Type: Expression(Integer)

axiom

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

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

(14)

Type: Expression(Integer)

axiom

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

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

(15)

Type: Expression(Integer)

axiom

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

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

(16)

Type: Expression(Integer)

axiom

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

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

(17)

Type: Expression(Integer)

axiom

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

$\label{eq18}\begin{array}{@{}l} \displaystyle \left[{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{1}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{1}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{1}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{1}}}}, \: \right. \ \ \displaystyle \left.{{{u^{2}}\ {v^{2}}\ {y_{2, \: 2}^{2}}}+{{u^{2}}\ {v^{1}}\ {y_{2, \: 1}^{2}}}+{{u^{1}}\ {v^{2}}\ {y_{1, \: 2}^{2}}}+{{u^{1}}\ {v^{1}}\ {y_{1, \: 1}^{2}}}}\right]$

(18)

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

axiom

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

Edit detail for SandBoxFrobeniusAlgebra revision 2 of 26