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last edited 13 years ago by Bill Page |
Edit detail for SandBox Grassmann Algebra Is Frobenius In Many Ways revision 4 of 8
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Editor: Bill Page
Time: 2011/04/05 15:36:06 GMT-7 |
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| Note: Frobenius pairing | ||
changed: -YU := reindex(reindex(U,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-U*Y; YU := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg changed: -and look for all associative scalar products $U = U(Y)$ or we -may consider an scalar product U as given, and look for all -algebras $Y=Y(U)$ such that the scalar product is associative. and look for all associative scalar products $U = U(Y)$ added: Alternatively we may consider removed: -K := jacobian(ravel(YU),concat(map(variables,ravel(Y)))::List Symbol); ---yy := transpose matrix [concat(map(variables,ravel(Y)))::List Symbol]; ---K::OutputForm * yy::OutputForm = 0 -\end{axiom} -The matrix 'K' transforms the coefficients of the tensor $Y$ -into coefficients of the tensor $\Phi$. We are looking for -coefficients of the tensor $U$ such that 'K' transforms the -tensor $Y$ into $\Phi=0$ for any $Y$. - -A necessary condition for the equation to have a non-trivial -solution is that the matrix 'K' be degenerate. - -Consider the determinant of the matrix 'K' above:: - - !\begin{axiom} - Kd := factor(determinant(K)::DMP(concat map(variables,ravel(U)),FRAC INT)) - \end{axiom} - -The scalar product must also be non-degenerate:: - - !\begin{axiom} - Ud:DMP(concat map(variables,ravel(U)),FRAC INT) := determinant [[U[i,j] for j in 1..dim] for i in 1..dim] - \end{axiom} - -The basis of the null space of the 'K' matrix:: - - !\begin{axiom} - YUS:T := reindex(reindex(US,[2,1])*reindex(Y,[1,3,2]),[3,2,1])-US*Y - KS := jacobian(ravel(YUS),concat(map(variables,ravel(Y)))::List Symbol); - NS:=nullSpace(KS) - SS:=map((x,y)+->x=y,concat map(variables,ravel Y), - entries reduce(+,[p[i]*NS.i for i in 1..#NS])) - YS:T := unravel(map(x+->subst(x,SS),ravel Y)) - \end{axiom} - -This defines a family of pre-Frobenius algebras:: - - !\begin{axiom} - test(unravel(map(x+->subst(x,SS),ravel YUS))$T=0*YU) - \end{axiom} - -Alternatively we may consider -\begin{axiom} changed: ---uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol]; ---J::OutputForm * uu::OutputForm = 0 uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol]; J::OutputForm * uu::OutputForm = 0 nrows(J) ncols(J) changed: -A necessary condition for the equation to have a non-trivial -solution is that all 70 of the 4x4 sub-matrices of 'J' are -degenerate. To this end we can form the polynomial ideal of -the determinants of these sub-matrices:: - - !\begin{axiom} - JP:=ideal concat concat concat - [[[[ determinant( - matrix([row(J,i1),row(J,i2),row(J,i3),row(J,i4)])) - for i4 in (i3+1)..maxRowIndex(J) ] - for i3 in (i2+1)..(maxRowIndex(J)-1) ] - for i2 in (i1+1)..(maxRowIndex(J)-2) ] - for i1 in minRowIndex(J)..(maxRowIndex(J)-3) ]; - #generators(%) - \end{axiom} - If the null space of the 'J' matrix is not empty we can use the basis to find all non-trivial solutions for U: \begin{axiom} NJ:=nullSpace(J) SS:=map((x,y)+->x=y,concat map(variables,ravel U), entries reduce(+,[p[i]*NJ.i for i in 1..#NJ])) Ug:T := unravel(map(x+->subst(x,SS),ravel U)) \end{axiom} This defines a family of pre-Frobenius algebras: \begin{axiom} test(unravel(map(x+->subst(x,SS),ravel YU))$T=0*YU) \end{axiom} The scalar product must be non-degenerate: \begin{axiom} Ud := determinant [[Ug[i,j] for j in 1..dim] for i in 1..dim] \end{axiom}
-dimensional vector space representing Grassmann algebra with 
generators
An algebra is represented by a (2,1)-tensor
viewed as a linear operator with two inputs 
and one
output 
. For example:
axiom
n:=2
 | (1) |
Type: PositiveInteger?
axiom
dim:=2^n
 | (2) |
Type: PositiveInteger?
axiom
T:=CartesianTensor(1,dim, FRAC POLY INT)
 | (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}}](images/6312196184808662070-16.0px.png "\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) |
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 ])](images/2192497802694895792-16.0px.png "\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]](images/8594787505525595887-16.0px.png "\label{eq6}\left[ 1, \:{g_{1}}, \:{g_{2}}, \:{{g_{1}}\ {g_{2}}}\right]") | (6) |
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](images/6814567651350444665-16.0px.png "\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) |
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](images/6426527865618376638-16.0px.png "\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) |
A scalar product is denoted by the (2,0)-tensor
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}}](images/4224929678563928732-16.0px.png "\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) |
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
 | (10) |
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}}](images/8043540721742164722-16.0px.png "\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) |
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.
Alternatively we may consider
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](images/4291865835332423217-16.0px.png "\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)
 | (13) |
Type: PositiveInteger?
axiom
ncols(J)
 | (14) |
Type: PositiveInteger?
The matrix J transforms the coefficients of the tensor 
into coefficients of the tensor 
. We are looking for
coefficients of the tensor 
such that J transforms the
tensor into 
for any .
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]](images/1355895520219202530-16.0px.png "\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]](images/9115553543507672162-16.0px.png "\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](images/2173993453491499047-16.0px.png "\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) |
This defines a family of pre-Frobenius algebras:
axiom
test(unravel(map(x+->subst(x,SS), ravel YU))$T=0*YU)
 | (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]
 | (19) |
Type: Fraction(Polynomial(Integer))