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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 12 years ago by Bill Page

Edit detail for SandBoxFrobeniusAlgebra revision 12 of 26

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Editor: Bill Page Time: 2011/02/14 15:12:09 GMT-8
Note: Jacobi identity

added:

The Jacobi identity requires the following tensor to be zero::

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

\begin{axiom}
BA := AA - reindex(contract(Y,1,Y,2),[3,1,4,2]); ravel(BA)
BB:=groebner(ravel(BA));
#BB
\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)
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))

$\label{eq2}\hbox{\axiomType{CartesianTensor}\ } (1, 2, \hbox{\axiomType{HomogeneousDistributedMultivariatePolynomial}\ } ([ 012 y 111, 012 y 112, 012 y 121, 012 y 122, 012 y 211, 012 y 212, 012 y 221, 012 y 222, 01 u 1, 01 u 2, 01 v 1, 01 v 2 ] , \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })))$

(2)

Type: Domain

axiom

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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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,HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(Integer)))

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

axiom

C:=groebner(ravel(K))

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

(9)

Type: List(HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(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(contract(Y,1,Y,2),[3,1,2,4])-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}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{1, \: 1}}\ {y_{1}^{1, \: 2}}}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}^2}-{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}+{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}, \: \right. \ \ \displaystyle \left.{-{{y_{1}^{2, \: 1}}^2}+{{y_{1}^{1, \: 1}}\ {y_{1}^{2, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}+{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \: \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, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{1}^{1, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{2}^{1, \: 2}}^2}+{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}}, \:{-{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}+{{y_{1}^{1, \: 2}}\ {y_{2}^{1, \: 2}}}}, \right. \ \ \displaystyle \left.\:{{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{1, \: 1}}\ {y_{2}^{2, \: 1}}}+{{y_{2}^{2, \: 1}}^2}-{{y_{2}^{1, \: 1}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 1}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{1, \: 2}}\ {y_{2}^{2, \: 1}}}-{{y_{2}^{1, \: 2}}\ {y_{2}^{2, \: 2}}}+{{y_{2}^{2, \: 1}}\ {y_{2}^{2, \: 2}}}}, \: \right. \ \ \displaystyle \left.{{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 1}}}-{{y_{1}^{2, \: 1}}\ {y_{2}^{2, \: 1}}}}, \:{{{y_{1}^{2, \: 2}}\ {y_{2}^{1, \: 2}}}-{{y_{1}^{2, \: 2}}\ {y_{2}^{2, \: 1}}}}\right]$

(14)

Type: List(HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(Integer)))

axiom

AB:=groebner(ravel(AA))

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

(15)

Type: List(HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(Integer)))

axiom

#AB

$\label{eq16}15$

(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)

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

(17)

Type: List(HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(Integer)))

axiom

BB:=groebner(ravel(BA));

Type: List(HomogeneousDistributedMultivariatePolynomial?([*012y111,*012y112,*012y121,*012y122,*012y211,*012y212,*012y221,*012y222,*01u1,*01u2,*01v1,*01v2],Fraction(Integer)))

axiom

#BB

$\label{eq18}24$

(18)

Type: PositiveInteger?