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# Edit detail for SandBoxFrobeniusAlgebra revision 1 of 26

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 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])$T \end{axiom} the tensor 'Y' operates on their tensor product \begin{axiom} UV:=product(U,V) YUV:=product(Y,UV) YUV.[1,1,1,1,2] YUV.[1,1,1,2,1] YUV.[1,1,2,1,1] YUV.[1,2,1,1,1] YUV.[2,1,1,1,1] Y*U*V \end{axiom}  An n-dimensional algebra is represented by a tensor viewed as an operator with two inputs i,j and one output k. axiom n:=2 (1) Type: PositiveInteger? axiom T:=CartesianTensor(1,n,EXPR INT) (2) Type: Domain axiom Y:=unravel(concat concat [[[script(y,[[i,j],[k]]) for i in 1..n] for j in 1..n] for k in 1..n] )$T (3)
Type: CartesianTensor?(1,2,Expression(Integer))

Given two vectors U and V

axiom
U:=unravel([script(u,[[],[i]]) for i in
1..n])$T (4) Type: CartesianTensor?(1,2,Expression(Integer)) axiom V:=unravel([script(v,[[],[i]]) for i in 1..n])$T (5)
Type: CartesianTensor?(1,2,Expression(Integer))

the tensor Y operates on their tensor product

axiom
UV:=product(U,V) (6)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
YUV:=product(Y,UV) (7)
Type: CartesianTensor?(1,2,Expression(Integer))
axiom
YUV.[1,1,1,1,2] (8)
Type: Expression(Integer)
axiom
YUV.[1,1,1,2,1] (9)
Type: Expression(Integer)
axiom
YUV.[1,1,2,1,1] (10)
Type: Expression(Integer)
axiom
YUV.[1,2,1,1,1] (11)
Type: Expression(Integer)
axiom
YUV.[2,1,1,1,1] (12)
Type: Expression(Integer)
axiom
Y*U*V (13)
Type: CartesianTensor?(1,2,Expression(Integer))