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	Frobenius Algebra, Vector Spaces and Polynomial Ideals
Octonion Algebra is Frobenius in Just One Way ←	↑	→ SandBox Quaternion Algebra is Frobenius in Many Ways

SandBox Grassmann Algebra Is Frobenius In Many Ways

Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 5 of 8

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Editor: Bill Page Time: 2011/04/05 16:12:02 GMT-7
Note: Frobenius co-pairing

removed:
-
-Alternatively we may consider

changed:
-coefficients of the tensor $Y$ such that 'J' transforms the
-tensor $U$ into $\Phi=0$ for any $U$.
the general linear family of tensors $U=U(Y,p_i)$ such that
'J' transforms $U$ into $\Phi=0$ for any such $U$.

added:
Co-pairing
\begin{axiom}
gU:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)
-- dimension
contract(contract(gU,1,Ug,1),1,2)
\end{axiom}

-dimensional vector space representing Grassmann algebra with generators

An algebra is represented by a (2,1)-tensor $Y=\{ {y^k}_{ij} \ i,j,k =1,2, ... dim \}$ viewed as a linear operator with two inputs and one output . For example:

axiom

n:=2

$\label{eq1}2$

(1)

Type: PositiveInteger?

axiom

dim:=2^n

$\label{eq2}4$

(2)

Type: PositiveInteger?

axiom

T:=CartesianTensor(1,dim,FRAC POLY INT)

$\label{eq3}\hbox{\axiomType{CartesianTensor}\ } (1, 4, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })))$

(3)

Type: Type

axiom

Y:T := unravel(concat concat
  [[[script(y,[[i,j],[k]])
    for i in 1..dim]
      for j in 1..dim]
        for k in 1..dim]
          )

$\label{eq4}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} {y_{1, \: 1}^{1}}&{y_{2, \: 1}^{1}}&{y_{3, \: 1}^{1}}&{y_{4, \: 1}^{1}} \ {y_{1, \: 2}^{1}}&{y_{2, \: 2}^{1}}&{y_{3, \: 2}^{1}}&{y_{4, \: 2}^{1}} \ {y_{1, \: 3}^{1}}&{y_{2, \: 3}^{1}}&{y_{3, \: 3}^{1}}&{y_{4, \: 3}^{1}} \ {y_{1, \: 4}^{1}}&{y_{2, \: 4}^{1}}&{y_{3, \: 4}^{1}}&{y_{4, \: 4}^{1}}$

(4)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

Generate structure constants for Grassmann Algebra

axiom

-- Construct a basis for the Grassmann algebra
GA:=AntiSymm(INT,[subscript(g,[i]) for i in 1..n])

$\label{eq5}\hbox{\axiomType{AntiSymm}\ } (\hbox{\axiomType{Integer}\ } , [ <em> 01 g 1, </em> 01 g 2 ])$

(5)

Type: Type

axiom

B:=[exp(reverse concat([0 for i in 1..n-length(x)],wholeRagits(x::RADIX(2))))$GA for x in 0..dim-1]

$\label{eq6}\left[ 1, \:{g_{1}}, \:{g_{2}}, \:{{g_{1}}\ {g_{2}}}\right]$

(6)

Type: List(AntiSymm?(Integer,[*01g1,*01g2]))

axiom

-- Compute the multiplication table
M:=matrix [[B.i * B.j for j in 1..dim] for i in 1..dim]

$\label{eq7}\left[ \begin{array}{cccc} 1 &{g_{1}}&{g_{2}}&{{g_{1}}\ {g_{2}}} \ {g_{1}}& 0 &{{g_{1}}\ {g_{2}}}& 0 \ {g_{2}}& -{{g_{1}}\ {g_{2}}}& 0 & 0 \ {{g_{1}}\ {g_{2}}}& 0 & 0 & 0$

(7)

Type: Matrix(AntiSymm?(Integer,[*01g1,*01g2]))

axiom

-- The structure constants of the algebra are given by the coefficients
-- of the polynomials in the multiplication table with respect to the basis
S(y)==map(x+->coefficient(x,y),M)

Type: Void

axiom

Yg:T:=unravel concat concat(map(S,B)::List List List FRAC POLY INT)

axiom

Compiling function S with type AntiSymm(Integer,[*01g1,*01g2]) -> 
      Matrix(Integer)

$\label{eq8}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0$

(8)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$

axiom

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

$\label{eq9}\left[ \begin{array}{cccc} {u^{1, \: 1}}&{u^{1, \: 2}}&{u^{1, \: 3}}&{u^{1, \: 4}} \ {u^{2, \: 1}}&{u^{2, \: 2}}&{u^{2, \: 3}}&{u^{2, \: 4}} \ {u^{3, \: 1}}&{u^{3, \: 2}}&{u^{3, \: 3}}&{u^{3, \: 4}} \ {u^{4, \: 1}}&{u^{4, \: 2}}&{u^{4, \: 3}}&{u^{4, \: 4}}$

(9)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

Definition 1

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

    Y   =   Y
     U     U

In other words, if the (3,0)-tensor:

    i  j  k   i  j  k   i  j  k
     \ | /     \/  /     \  \/
      \|/   =   \ /   -   \ /
       0         0         0

$\label{eq10} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(10)

(three-point function) is zero.

axiom

YU := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg

$\label{eq11}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ {{u^{2, \: 1}}-{u^{1, \: 2}}}&{u^{2, \: 2}}&{{u^{2, \: 3}}-{u^{1, \: 4}}}&{u^{2, \: 4}} \ {{u^{3, \: 1}}-{u^{1, \: 3}}}&{{u^{3, \: 2}}+{u^{1, \: 4}}}&{u^{3, \: 3}}&{u^{3, \: 4}} \ {{u^{4, \: 1}}-{u^{1, \: 4}}}&{u^{4, \: 2}}&{u^{4, \: 3}}&{u^{4, \: 4}}$

(11)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

Definition 2

An algebra with a non-degenerate associative scalar product is called pre-Frobenius.

We may consider the problem where multiplication Y is given, and look for all associative scalar products

This problem can be solved using linear algebra.

axiom

)expose MCALCFN

   MultiVariableCalculusFunctions is now explicitly exposed in frame 
      initial 
J := jacobian(ravel(YU),concat(map(variables,ravel(U)))::List Symbol);

Type: Matrix(Fraction(Polynomial(Integer)))

axiom

uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];

Type: Matrix(Polynomial(Integer))

axiom

J::OutputForm * uu::OutputForm = 0

$\label{eq12}\begin{array}{@{}l} \displaystyle {{\left[ \begin{array}{cccccccccccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & - 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0$

(12)

Type: Equation(OutputForm?)

axiom

nrows(J)

$\label{eq13}64$

(13)

Type: PositiveInteger?

axiom

ncols(J)

$\label{eq14}16$

(14)

Type: PositiveInteger?

The matrix J transforms the coefficients of the tensor into coefficients of the tensor $\Phi$ . We are looking for the general linear family of tensors such that J transforms into $\Phi=0$ for any such .

If the null space of the J matrix is not empty we can use the basis to find all non-trivial solutions for U:

axiom

NJ:=nullSpace(J)

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{\left[ 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}\right]$

(15)

Type: List(Vector(Fraction(Polynomial(Integer))))

axiom

SS:=map((x,y)+->x=y,concat map(variables,ravel U),
  entries reduce(+,[p[i]*NJ.i for i in 1..#NJ]))

$\label{eq16}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}={p_{1}}}, \:{{u^{1, \: 2}}={p_{2}}}, \:{{u^{1, \: 3}}={p_{3}}}, \:{{u^{1, \: 4}}={p_{4}}}, \:{{u^{2, \: 1}}={p_{2}}}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 2}}= 0}, \:{{u^{2, \: 3}}={p_{4}}}, \:{{u^{2, \: 4}}= 0}, \:{{u^{3, \: 1}}={p_{3}}}, \:{{u^{3, \: 2}}= -{p_{4}}}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 3}}= 0}, \:{{u^{3, \: 4}}= 0}, \:{{u^{4, \: 1}}={p_{4}}}, \:{{u^{4, \: 2}}= 0}, \:{{u^{4, \: 3}}= 0}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 4}}= 0}\right]$

(16)

Type: List(Equation(Fraction(Polynomial(Integer))))

axiom

Ug:T := unravel(map(x+->subst(x,SS),ravel U))

$\label{eq17}\left[ \begin{array}{cccc} {p_{1}}&{p_{2}}&{p_{3}}&{p_{4}} \ {p_{2}}& 0 &{p_{4}}& 0 \ {p_{3}}& -{p_{4}}& 0 & 0 \ {p_{4}}& 0 & 0 & 0$

(17)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

This defines a family of pre-Frobenius algebras:

axiom

test(unravel(map(x+->subst(x,SS),ravel YU))$T=0*YU)

$\label{eq18} \mbox{\rm true}$

(18)

Type: Boolean

The scalar product must be non-degenerate:

axiom

Ud := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]

$\label{eq19}-{{p_{4}}^4}$

(19)

Type: Fraction(Polynomial(Integer))

Co-pairing

axiom

gU:T:=unravel concat(transpose(1/Ud*adjoint([[Ug[i,j] for j in 1..dim] for i in 1..dim]).adjMat)::List List FRAC POLY INT)

$\label{eq20}\left[ \begin{array}{cccc} 0 & 0 & 0 &{1 \over{p_{4}}} \ 0 & 0 &{1 \over{p_{4}}}& -{{p_{3}}\over{{p_{4}}^2}} \ 0 & -{1 \over{p_{4}}}& 0 &{{p_{2}}\over{{p_{4}}^2}} \ {1 \over{p_{4}}}&{{p_{3}}\over{{p_{4}}^2}}& -{{p_{2}}\over{{p_{4}}^2}}& -{{p_{1}}\over{{p_{4}}^2}}$

(20)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))

axiom

-- dimension
contract(contract(gU,1,Ug,1),1,2)

$\label{eq21}4$

(21)

Type: CartesianTensor?(1,4,Fraction(Polynomial(Integer)))