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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 13 of 26

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Editor: Bill Page Time: 2011/02/14 15:37:10 GMT-8
Note: scalar product

changed:
-$Y=\{ {y_k}^{ji} \} \ i,j,k =1,2, ... n $
$Y=\{ {y_k}^{ji} \ i,j,k =1,2, ... n \}$

changed:
---T:=CartesianTensor(1,n,DEXPR INT)
-T:=CartesianTensor(1,n,HDMP(concat[concat concat
-  [[[script(y,[[k],[j,i]])
-    for i in 1..n]
-      for j in 1..n]
-        for k in 1..n],
-          [script(u,[[i]]) for i in 1..n],
-            [script(v,[[i]]) for i in 1..n] ],FRAC INT))
T:=CartesianTensor(1,n,DEXPR INT)
--T:=CartesianTensor(1,n,HDMP(concat[concat concat
--  [[[script(y,[[k],[j,i]])
--    for i in 1..n]
--      for j in 1..n]
--        for k in 1..n],
--          [script(u,[[i]]) for i in 1..n],
--            [script(v,[[i]]) for i in 1..n] ],FRAC INT))

changed:
-test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])=reindex(contract(Y,1,Y,2),[3,1,2,4]))
-AA := reindex(contract(Y,1,Y,2),[3,1,2,4])-Y*Y; ravel(AA)
test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) = reindex(contract(Y,1,Y,2),[3,1,2,4]))
AA := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y; ravel(AA)

changed:
-
A scalar product is denoted by $U = \{ u^{ij} \}$
\begin{axiom}
U:=unravel(concat
  [[script(u,[[],[j,i]])
    for i in 1..n]
      for j in 1..n]
        )$T
\end{axiom}
We say that the scalar product is "associative" if the following tensor equation holds::

  Y I = I Y
   U     U

\begin{axiom}
UA := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y; ravel(UA)
\end{axiom}

An n-dimensional algebra is represented by a (1,2)-tensor $Y=\{ {y_k}^{ji} \ i,j,k =1,2, ... n \}$ viewed as an operator with two inputs i,j and one output k. For example in 2 dimensions

axiom

)library DEXPR

   DistributedExpression is now explicitly exposed in frame initial
   DistributedExpression will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/DEXPR.NRLIB/DEXPR
n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

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

$\label{eq2}\hbox{\axiomType{CartesianTensor}\ } (1, 2, \hbox{\axiomType{DistributedExpression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Domain

axiom

--T:=CartesianTensor(1,n,HDMP(concat[concat concat
--  [[[script(y,[[k],[j,i]])
--    for i in 1..n]
--      for j in 1..n]
--        for k in 1..n],
--          [script(u,[[i]]) for i in 1..n],
--            [script(v,[[i]]) for i in 1..n] ],FRAC
INT))
Y:=unravel(concat concat
  [[[script(y,[[k],[j,i]])
    for i in 1..n]
      for j in 1..n]
        for k in 1..n]
          )$T

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} {y_{1}^{1, \: 1}}&{y_{1}^{1, \: 2}} \ {y_{1}^{2, \: 1}}&{y_{1}^{2, \: 2}}$

(3)

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

Given two vectors $U=\{ u_i \}$ and $V=\{ v_j \}$

axiom

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

$\label{eq4}\left[{u_{1}}, \:{u_{2}}\right]$

(4)

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

axiom

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

$\label{eq5}\left[{v_{1}}, \:{v_{2}}\right]$

(5)

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

the tensor Y operates on their tensor product to yield a vector $W=\{ w_k = {y_k}^{ji} u_i v_j \}$

axiom

W:=contract(contract(Y,3,product(U,V),1),2,3)

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{{{y_{1}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{1}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{1}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{2}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{2}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}\right]$

(6)

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

Diagram:

  U   V
  2i  3j
   \ /
    |
    1k
    W

or in a more convenient notation:

axiom

W:=(Y*U)*V

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{{{y_{1}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{1}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{1}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{1}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {u_{1}}\ {v_{1}}}+{{y_{2}^{1, \: 2}}\ {u_{2}}\ {v_{1}}}+{{y_{2}^{2, \: 1}}\ {u_{1}}\ {v_{2}}}+{{y_{2}^{2, \: 2}}\ {u_{2}}\ {v_{2}}}}\right]$

(7)

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

The algebra Y is commutative if the following tensor (the commutator) is zero

axiom

K:=Y-reindex(Y,[1,3,2])

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cc} 0 &{{y_{1}^{1, \: 2}}-{y_{1}^{2, \: 1}}} \ {-{y_{1}^{1, \: 2}}+{y_{1}^{2, \: 1}}}& 0$

(8)

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

A basis for the ideal defined by the coefficients of the commutator is given by:

axiom

C:=groebner(ravel(K))

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

(9)

Type: List(Polynomial(Integer))

An algebra is associative if:

  Y I  =  I Y
   Y       Y

  Note: right figure is mirror image of left!

  2  3 6   2 5  6      2  3  4
   \/ /     \ \/        \ | /
    \/   =   \/    =     \|/
     \       /            |
      4     1             1

In other words an algebra is associative if and only if the following (3,1)-tensor $A=\{ {a_s}^{kji} = {y_s}^{kr} {y_r}^{ji} - {y_r}^{kj} {y_s}^{ri} \}$ is zero.

axiom

test(Y*Y = contract(product(Y,Y),3,4))

$\label{eq10} \mbox{\rm true}$

(10)

Type: Boolean

axiom

test(Y*Y = contract(Y,3,Y,1))

$\label{eq11} \mbox{\rm true}$

(11)

Type: Boolean

axiom

test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) =
reindex(contract(product(Y,Y),1,5),[3,1,2,4]))

$\label{eq12} \mbox{\rm true}$

(12)

Type: Boolean

axiom

test(reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2]) = reindex(contract(Y,1,Y,2),[3,1,2,4]))

$\label{eq13} \mbox{\rm true}$

(13)

Type: Boolean

axiom

AA := reindex(reindex(Y,[1,3,2])*reindex(Y,[1,3,2]),[1,4,3,2])-Y*Y;
ravel(AA)

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{y_{1}^{1, \: 2}}^2}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \:{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^2}+{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}\ {y_{1}^{2, \: 2}}}}, \:{-{{y_{2}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{1, \: 2}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \:{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}-{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}+{{y_{2}^{2, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{2}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{y_{2}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}\right]$

(14)

Type: List(DistributedExpression?(Integer))

axiom

AB:=groebner(ravel(AA))

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{{{\left({y_{2}^{2, \: 1}}-{y_{2}^{1, \: 2}}\right)}\ {y_{2}^{2, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{1, \: 2}}^2}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{2, \: 1}}^2}-{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{\left({{y_{2}^{1, \: 2}}^2}-{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}\right)}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}^3}+{{y_{1}^{1, \: 1}}\ {{y_{2}^{1, \: 2}}^2}}}, \: \right. \ \ \displaystyle \left.{{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}}, \: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{y_{1}^{2, \: 1}}\ {{y_{2}^{1, \: 2}}^2}}+{{\left(-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}\right)}\ {y_{2}^{1, \: 2}}}+ \ \ \displaystyle {{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}$

(15)

Type: List(Polynomial(Integer))

axiom

#AB

$\label{eq16}16$

(16)

Type: PositiveInteger?

The Jacobi identity requires the following tensor to be zero:

  2    3 6   2 5    6   2 6  3   
   \  / /     \ \  /     \ \/   
    \/ /       \ \/       \/\   
     \/    -    \/    -    \/   
      \         /           \   
       4       1             4

axiom

BA := AA - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(BA)

$\label{eq17}\begin{array}{@{}l} \displaystyle \left[{-{{y_{1}^{1, \: 1}}^2}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \right. \ \ \displaystyle \left.\:{-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \right. \ \ \displaystyle \left.\: \right. \ \ \displaystyle \left.{ \begin{array}{@{}l} \displaystyle {{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{1}^{2, \: 1}}}-{{y_{1}^{2, \: 1}}^2}+{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}- \ \ \displaystyle {2 \ {y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}$

(17)

Type: List(DistributedExpression?(Integer))

axiom

BB:=groebner(ravel(BA));

Type: List(Polynomial(Integer))

axiom

#BB

$\label{eq18}26$

(18)

Type: PositiveInteger?

A scalar product is denoted by $U = \{ u^{ij} \}$

axiom

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

$\label{eq19}\left[ \begin{array}{cc} {u^{1, \: 1}}&{u^{1, \: 2}} \ {u^{2, \: 1}}&{u^{2, \: 2}}$

(19)

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

We say that the scalar product is "associative" if the following tensor equation holds:

  Y I = I Y
   U     U

axiom

UA := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y; ravel(UA)

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

(20)

Type: List(DistributedExpression?(Integer))