help links subscribe changes refresh edit

	Frobenius Algebra, Vector Spaces and Polynomial Ideals
SandBox Grassmann Algebra Is Frobenius In Many Ways ←	↑	→ SandBox Sedenion Algebra is Frobenius In Just One Way

SandBox Quaternion Algebra is Frobenius in Many Ways

last edited 11 years ago by Bill Page

Edit detail for SandBox Quaternion Algebra is Frobenius in Many Ways revision 3 of 32

	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 27 28 29 30 31 32
Editor: page Time: 2011/04/08 00:04:06 GMT-7
Note: S3 permutations

changed:
-  See::
-
    Zbigniew Oziewicz, Gregory Peter Wene
    (26 Mar 2011)

removed:
-    (Submitted on 26 Mar 2011)
-    Zbigniew Oziewicz (UNAM), Gregory Peter Wene (UTSA)

4-dimensional vector space representing Quaternion algebra

axiom

dim:=4

$\label{eq1}4$

(1)

Type: PositiveInteger?

axiom

T := CartesianTensor(1,dim,EXPR INT)

$\label{eq2}\hbox{\axiomType{CartesianTensor}\ } (1, 4, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))$

(2)

Type: Type

axiom

X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]

$\label{eq3}\left[{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 1, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: 1 \right]}\right]$

(3)

Type: List(CartesianTensor?(1,4,Expression(Integer)))

axiom

X(1),X(2)

$\label{eq4}\left[{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 1, \: 0, \: 0 \right]}\right]$

(4)

Type: Tuple(CartesianTensor?(1,4,Expression(Integer)))

Generate structure constants for Quaternion Algebra

axiom

B:=map(x+->quatern(x.1,x.2,x.3,x.4),1$SQMATRIX(4,FRAC INT)::List List FRAC INT)

$\label{eq5}\left[ 1, \: i , \: j , \: k \right]$

(5)

Type: List(Quaternion(Fraction(Integer)))

axiom

M:=matrix [[B.i*B.j for j in 1..4] for i in 1..4]

$\label{eq6}\left[ \begin{array}{cccc} 1 & i & j & k \ i & - 1 & k & - j \ j & - k & - 1 & i \ k & j & - i & - 1$

(6)

Type: Matrix(Quaternion(Fraction(Integer)))

axiom

S(y)==map(x+->(x*inv(y)=1 or x*inv(y)=-1 => x*inv(y);0),M)

Type: Void

axiom

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

axiom

Compiling function S with type Quaternion(Fraction(Integer)) -> 
      Matrix(Quaternion(Fraction(Integer)))

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 \ 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & - 1$

(7)

Type: CartesianTensor?(1,4,Expression(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]
        )

$\label{eq8}\left[ \begin{array}{cccc} {u^{1, \: 1}}&{u^{1, \: 2}}&{u^{1, \: 3}}&{u^{1, \: 4}} \ {u^{2, \: 1}}&{u^{2, \: 2}}&{u^{2, \: 3}}&{u^{2, \: 4}} \ {u^{3, \: 1}}&{u^{3, \: 2}}&{u^{3, \: 3}}&{u^{3, \: 4}} \ {u^{4, \: 1}}&{u^{4, \: 2}}&{u^{4, \: 3}}&{u^{4, \: 4}}$

(8)

Type: CartesianTensor?(1,4,Expression(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

$\label{eq9} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}$

(9)

(three-point function) is zero.

axiom

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

$\label{eq10}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 & 0 & 0 & 0 \ {{u^{2, \: 1}}-{u^{1, \: 2}}}&{{u^{2, \: 2}}+{u^{1, \: 1}}}&{{u^{2, \: 3}}-{u^{1, \: 4}}}&{{u^{2, \: 4}}+{u^{1, \: 3}}} \ {{u^{3, \: 1}}-{u^{1, \: 3}}}&{{u^{3, \: 2}}+{u^{1, \: 4}}}&{{u^{3, \: 3}}+{u^{1, \: 1}}}&{{u^{3, \: 4}}-{u^{1, \: 2}}} \ {{u^{4, \: 1}}-{u^{1, \: 4}}}&{{u^{4, \: 2}}-{u^{1, \: 3}}}&{{u^{4, \: 3}}+{u^{1, \: 2}}}&{{u^{4, \: 4}}+{u^{1, \: 1}}}$

(10)

Type: CartesianTensor?(1,4,Expression(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 ω,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

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

Type: Matrix(Polynomial(Integer))

axiom

J::OutputForm * uu::OutputForm = 0

$\label{eq11}\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 \ 1 & 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 & 1 & 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 \ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & - 1 & 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 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 1 & 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 \ - 1 & 0 & 0 & 0 & 0 & - 1 & 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 & - 1 & 0 & 0 & 0 & 0 & - 1 & 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 & - 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 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & - 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 & 1 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 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 & - 1 & 0 & 0 & - 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 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 & 1 & 0 & 0 & 0 & 0 & - 1 \ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 \ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 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 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & - 1 & 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 & 1 & 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 & 1 & 0 & 0 & 0 & 0 & - 1 & 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 & 1 & 0 & 0 & 0 & 0 & - 1 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \ 0 & 0 & 0 & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 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 \ - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \ 0 & - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 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 & 1 & 0 & 0 & 0$

(11)

Type: Equation(OutputForm?)

axiom

nrows(J)

$\label{eq12}64$

(12)

Type: PositiveInteger?

axiom

ncols(J)

$\label{eq13}16$

(13)

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)

$\label{eq14}\begin{array}{@{}l} \displaystyle \left[{\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: - 1, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: - 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ - 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}\right]$

(14)

Type: List(Vector(Expression(Integer)))

axiom

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

$\label{eq15}\begin{array}{@{}l} \displaystyle \left[{{u^{1, \: 1}}= -{p_{4}}}, \:{{u^{1, \: 2}}= -{p_{3}}}, \:{{u^{1, \: 3}}={p_{2}}}, \:{{u^{1, \: 4}}={p_{1}}}, \: \right. \ \ \displaystyle \left.{{u^{2, \: 1}}= -{p_{3}}}, \:{{u^{2, \: 2}}={p_{4}}}, \:{{u^{2, \: 3}}={p_{1}}}, \:{{u^{2, \: 4}}= -{p_{2}}}, \: \right. \ \ \displaystyle \left.{{u^{3, \: 1}}={p_{2}}}, \:{{u^{3, \: 2}}= -{p_{1}}}, \:{{u^{3, \: 3}}={p_{4}}}, \:{{u^{3, \: 4}}= -{p_{3}}}, \: \right. \ \ \displaystyle \left.{{u^{4, \: 1}}={p_{1}}}, \:{{u^{4, \: 2}}={p_{2}}}, \:{{u^{4, \: 3}}={p_{3}}}, \:{{u^{4, \: 4}}={p_{4}}}\right]$

(15)

Type: List(Equation(Expression(Integer)))

axiom

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

$\label{eq16}\left[ \begin{array}{cccc} -{p_{4}}& -{p_{3}}&{p_{2}}&{p_{1}} \ -{p_{3}}&{p_{4}}&{p_{1}}& -{p_{2}} \ {p_{2}}& -{p_{1}}&{p_{4}}& -{p_{3}} \ {p_{1}}&{p_{2}}&{p_{3}}&{p_{4}}$

(16)

Type: CartesianTensor?(1,4,Expression(Integer))

This defines a family of pre-Frobenius algebras:

axiom

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

$\label{eq17} \mbox{\rm true}$

(17)

Type: Boolean

The scalar product must be non-degenerate:

axiom

Ud:DMP([p[i] for i in 1..#NJ],INT) := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim]

$\label{eq18}\begin{array}{@{}l} \displaystyle -{{p_{1}}^4}-{2 \ {{p_{1}}^2}\ {{p_{2}}^2}}-{2 \ {{p_{1}}^2}\ {{p_{3}}^2}}-{2 \ {{p_{1}}^2}\ {{p_{4}}^2}}-{{p_{2}}^4}- \ \ \displaystyle {2 \ {{p_{2}}^2}\ {{p_{3}}^2}}-{2 \ {{p_{2}}^2}\ {{p_{4}}^2}}-{{p_{3}}^4}-{2 \ {{p_{3}}^2}\ {{p_{4}}^2}}-{{p_{4}}^4}$

(18)

Type: DistributedMultivariatePolynomial?([*01p1,*01p2,*01p3,*01p4],Integer)

axiom

factor Ud

$\label{eq19}-{{\left({{p_{1}}^2}+{{p_{2}}^2}+{{p_{3}}^2}+{{p_{4}}^2}\right)}^2}$

(19)

Type: Factored(DistributedMultivariatePolynomial?([*01p1,*01p2,*01p3,*01p4],Integer))

Definition 3

Co-pairing

axiom

Ωg: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)

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

(20)

Type: CartesianTensor?(1,4,Expression(Integer))

dimension
Ω
U

axiom

contract(contract(Ωg,1,Ug,1),1,2)

$\label{eq21}4$

(21)

Type: CartesianTensor?(1,4,Expression(Integer))

Definition 4

Co-multiplication

axiom

λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

-- just for display
reindex(λg,[3,1,2])

$\label{eq22}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} -{{p_{4}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}& -{{p_{3}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{2}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{1}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}} \ -{{p_{3}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{4}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{1}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}& -{{p_{2}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}} \ {{p_{2}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}& -{{p_{1}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{4}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}& -{{p_{3}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}} \ {{p_{1}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{2}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{3}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}&{{p_{4}}\over{{{p_{4}}^2}+{{p_{3}}^2}+{{p_{2}}^2}+{{p_{1}}^2}}}$

(22)

Type: CartesianTensor?(1,4,Expression(Integer))

i  
λ=Ω

axiom

test(λg*X(1)=Ωg)

$\label{eq23} \mbox{\rm true}$

(23)

Type: Boolean

Definition 5

Co-unit

  i 
  U

axiom

ιg:=X(1)*Ug

$\label{eq24}\left[ -{p_{4}}, \: -{p_{3}}, \:{p_{2}}, \:{p_{1}}\right]$

(24)

Type: CartesianTensor?(1,4,Expression(Integer))

Y=U
ι

axiom

test(ιg * Yg = Ug)

$\label{eq25} \mbox{\rm true}$

(25)

Type: Boolean

For example:

axiom

Ug0:T:=unravel eval(ravel Ug,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])

$\label{eq26}\left[ \begin{array}{cccc} - 1 & 0 & 0 & 1 \ 0 & 1 & 1 & 0 \ 0 & - 1 & 1 & 0 \ 1 & 0 & 0 & 1$

(26)

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1])

$\label{eq27}\left[ \begin{array}{cccc} -{1 \over 2}& 0 & 0 &{1 \over 2} \ 0 &{1 \over 2}&{1 \over 2}& 0 \ 0 & -{1 \over 2}&{1 \over 2}& 0 \ {1 \over 2}& 0 & 0 &{1 \over 2}$

(27)

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

λg0:T:=unravel eval(ravel λg,[p[1]=1,p[2]=0,p[3]=0,p[4]=1]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

reindex(λg0,[3,1,2])

$\label{eq28}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} -{1 \over 2}& 0 & 0 &{1 \over 2} \ 0 &{1 \over 2}&{1 \over 2}& 0 \ 0 & -{1 \over 2}&{1 \over 2}& 0 \ {1 \over 2}& 0 & 0 &{1 \over 2}$

(28)

Type: CartesianTensor?(1,4,Expression(Integer))

-permuted Frobenius Algebras

Zbigniew Oziewicz, Gregory Peter Wene (26 Mar 2011) http://arxiv.org/abs/1103.5113

axiom

test( Yg = reindex(reindex(  reindex(Ug*Yg,[1,2,3]),  [2,3,1])*Ωg,[3,1,2]) )

$\label{eq29} \mbox{\rm true}$

(29)

Type: Boolean

axiom

Yg213 := reindex(reindex(  reindex(Ug*Yg,[2,1,3]),  [2,3,1])*Ωg,[3,1,2]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

ω213  := reindex(reindex(U,[2,1])*reindex(Yg213,[1,3,2]),[3,2,1])-U*Yg213;

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

J213  := jacobian(ravel ω213,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

NJ213 := nullSpace(J213)

$\label{eq30}\left[ \right]$

(30)

Type: List(Vector(Expression(Integer)))

axiom

-- opposite algebra
Yg132 := reindex(reindex(  reindex(Ug*Yg,[1,3,2]),  [2,3,1])*Ωg,[3,1,2]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

ω132  := reindex(reindex(U,[2,1])*reindex(Yg132,[1,3,2]),[3,2,1])-U*Yg132;

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

J132  := jacobian(ravel ω132,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

NJ132 := nullSpace(J132)

$\label{eq31}\begin{array}{@{}l} \displaystyle \left[{\left[ 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 0, \: - 1, \: 0, \: 0, \: 0, \: 0, \: - 1, \: - 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ 0, \: 1, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 0, \: 0, \: - 1, \: 0, \: 0, \: 1, \: 0 \right]}, \: \right. \ \ \displaystyle \left.{\left[ - 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1, \: 0, \: 0, \: 0, \: 0, \: 1 \right]}\right]$

(31)

Type: List(Vector(Expression(Integer)))

axiom

Yg321 := reindex(reindex(  reindex(Ug*Yg,[3,2,1]),  [2,3,1])*Ωg,[3,1,2]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

ω321  := reindex(reindex(U,[2,1])*reindex(Yg321,[1,3,2]),[3,2,1])-U*Yg321;

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

J321  := jacobian(ravel ω321,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

NJ321 := nullSpace(J321)

$\label{eq32}\left[ \right]$

(32)

Type: List(Vector(Expression(Integer)))

axiom

Yg312 := reindex(reindex(  reindex(Ug*Yg,[3,1,2]),  [2,3,1])*Ωg,[3,1,2]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

ω312  := reindex(reindex(U,[2,1])*reindex(Yg312,[1,3,2]),[3,2,1])-U*Yg312;

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

J312  := jacobian(ravel ω312,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

NJ312 := nullSpace(J312)

$\label{eq33}\left[ \right]$

(33)

Type: List(Vector(Expression(Integer)))

axiom

Yg231 := reindex(reindex(  reindex(Ug*Yg,[2,3,1]),  [2,3,1])*Ωg,[3,1,2]);

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

ω231  := reindex(reindex(U,[2,1])*reindex(Yg231,[1,3,2]),[3,2,1])-U*Yg231;

Type: CartesianTensor?(1,4,Expression(Integer))

axiom

J231  := jacobian(ravel ω231,concat(map(variables,ravel U))::List Symbol);

Type: Matrix(Expression(Integer))

axiom

NJ231 := nullSpace(J231)

$\label{eq34}\left[ \right]$

(34)

Type: List(Vector(Expression(Integer)))