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Edit detail for SandBox Sedenion Algebra is Frobenius In Just One Way revision 1 of 5
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changed: - Sedenion Algebra is Frobenius in just one way! 16-dimensional vector space representing Sedenion Algebra Ref: http://en.wikipedia.org/wiki/Sedenion \begin{axiom} )set output tex off )set output algebra on \end{axiom} \begin{axiom} dim:=16 R ==> EXPR INT T ==> CartesianTensor(1,dim,EXPR INT) X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim] X(1),X(2) \end{axiom} Generate structure constants for Sedenion Algebra (the Caley-Dickson way) Ref: http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction \begin{axiom} O ==> Octonion R SED ==> DirectProduct(2,O) --B0:=map(x+->seden(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8,x.9,x.10,x.11,x.12,x.13,x.14,x.15,x.16),1$SQMATRIX(dim,R)::List List R) pair(x:O,y:O):SED==directProduct vector [x,y] caleyOne:=pair(1,0) B:=map(x+->pair(octon(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8),octon(x.9,x.10,x.11,x.12,x.13,x.14,x.15,x.16)),1$SQMATRIX(dim,R)::List List R) \end{axiom} \begin{axiom} caleyMul(x:SED,y:SED):SED == pair((x.1)*(y.1) - conjugate(y.2)*(x.2), (y.2)*(x.1) + (x.2)*conjugate(y.1)) caleyMul(caleyOne,caleyOne) --M0:=matrix [[B0.i*B0.j for j in 1..dim] for i in 1..dim] M:=matrix [[caleyMul(B.i,B.j) for j in 1..dim] for i in 1..dim] \end{axiom} \begin{axiom} caleyConj(x:SED):SED == pair(conjugate(x.1), -x.2) caleyInv(x:SED):SED == inv(caleyMul(caleyConj x,x).1) * caleyConj(x) --S0(y)==map(x+->(x*inv(y)=1 or x*inv(y)=-1 => x*inv(y);0),M0) --S0(B0.1) S(y)==map(x+->(caleyMul(x,caleyInv y)=caleyOne => 1;caleyMul(x,caleyInv y)=-caleyOne => -1;0),M) S(B.1) --Yg0:T:=unravel concat concat(map(S0,B0)::List List List R); Yg:T:=unravel concat concat(map(S,B)::List List List R) test(Yg0=Yg) \end{axiom} A scalar product is denoted by the (2,0)-tensor $U = \{ u_{ij} \}$ \begin{axiom} U:T := unravel(concat [[script(u,[[],[j,i]]) for i in 1..dim] for j in 1..dim] ); \end{axiom} 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 \begin{equation} \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \} \end{equation} (three-point function) is zero. \begin{axiom} ω := reindex(reindex(U,[2,1])*reindex(Yg,[1,3,2]),[3,2,1])-U*Yg; \end{axiom} 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 $U = U(Y)$ This problem can be solved using linear algebra. \begin{axiom} )expose MCALCFN J := jacobian(ravel ω,concat(map(variables,ravel U))::List Symbol); uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol]; J::OutputForm * uu::OutputForm = 0; nrows(J) ncols(J) \end{axiom} The matrix 'J' transforms the coefficients of the tensor $U$ into coefficients of the tensor $\Phi$. We are looking for the general linear family of tensors $U=U(Y,p_i)$ such that 'J' transforms $U$ into $\Phi=0$ for any such $U$. 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 ω))$T=0*ω) \end{axiom} The scalar product must be non-degenerate: \begin{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] factor Ud \end{axiom} Definition 3 Co-pairing \begin{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) \end{axiom} <center><pre> dimension Ω U </pre></center> \begin{axiom} contract(contract(Ωg,1,Ug,1),1,2) \end{axiom} Definition 4 Co-multiplication \begin{axiom} λg:=reindex(contract(contract(Ug*Yg,1,Ωg,1),1,Ωg,1),[2,3,1]); -- just for display reindex(λg,[3,1,2]) \end{axiom} <center><pre> i λ=Ω </pre></center> \begin{axiom} test(λg*X(1)=Ωg) \end{axiom} Definition 5 <center>Co-unit<pre> i U </pre></center> \begin{axiom} ιg:=X(1)*Ug \end{axiom} <center><pre> Y=U ι </pre></center> \begin{axiom} test(ιg * Yg = Ug) \end{axiom} For example: \begin{axiom} Ug0:T:=unravel eval(ravel Ug,[p[1]=1]) Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1]) λg0:T:=unravel eval(ravel λg,[p[1]=1]); reindex(λg0,[3,1,2]) \end{axiom}
Sedenion Algebra is Frobenius in just one way!
16-dimensional vector space representing Sedenion Algebra
Ref: http://en.wikipedia.org/wiki/Sedenion
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(1) -> )set output tex off
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)set output algebra on
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dim:=16
(1) 16
Type: PositiveInteger?
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R ==> EXPR INT
Type: Void
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T ==> CartesianTensor(1,dim, EXPR INT)
Type: Void
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X:List T := [unravel [(i=j => 1;0) for j in 1..dim] for i in 1..dim]
(4) [[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, 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, 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, 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]]
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X(1),X(2)
(5) [[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]]
Generate structure constants for Sedenion Algebra (the Caley-Dickson way)
Ref: http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
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O ==> Octonion R
Type: Void
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SED ==> DirectProduct(2,O)
Type: Void
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--B0:=map(x+->seden(x.1,x.2, x.3, x.4, x.5, x.6, x.7, x.8, x.9, x.10, x.11, x.12, x.13, x.14, x.15, x.16), 1$SQMATRIX(dim, R)::List List R) pair(x:O, y:O):SED==directProduct vector [x, y]
Function declaration pair : (Octonion(Expression(Integer)),Octonion (Expression(Integer))) -> DirectProduct(2, Octonion(Expression( Integer))) has been added to workspace.
Type: Void
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caleyOne:=pair(1,0)
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Compiling function pair with type (Octonion(Expression(Integer)),Octonion(Expression(Integer))) -> DirectProduct(2, Octonion( Expression(Integer)))
(9) [1,0]
Type: DirectProduct?(2,
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B:=map(x+->pair(octon(x.1,x.2, x.3, x.4, x.5, x.6, x.7, x.8), octon(x.9, x.10, x.11, x.12, x.13, x.14, x.15, x.16)), 1$SQMATRIX(dim, R)::List List R)
(10) [[1,0], [i, 0], [j, 0], [k, 0], [E, 0], [I, 0], [J, 0], [K, 0], [0, 1], [0, i], [0, j], [0, k], [0, E], [0, I], [0, J], [0, K]]
Type: List(DirectProduct?(2,
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caleyMul(x:SED,y:SED):SED == pair((x.1)*(y.1) - conjugate(y.2)*(x.2), (y.2)*(x.1) + (x.2)*conjugate(y.1))
Function declaration caleyMul : (DirectProduct(2,Octonion(Expression (Integer))), DirectProduct(2, Octonion(Expression(Integer)))) -> DirectProduct(2, Octonion(Expression(Integer))) has been added to workspace.
Type: Void
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caleyMul(caleyOne,caleyOne)
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Compiling function caleyMul with type (DirectProduct(2,Octonion( Expression(Integer))), DirectProduct(2, Octonion(Expression( Integer)))) -> DirectProduct(2, Octonion(Expression(Integer)))
(12) [1,0]
Type: DirectProduct?(2,
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--M0:=matrix [[B0.i*B0.j for j in 1..dim] for i in 1..dim] M:=matrix [[caleyMul(B.i,B.j) for j in 1..dim] for i in 1..dim]
(13) [ [[1,0], [i, 0], [j, 0], [k, 0], [E, 0], [I, 0], [J, 0], [K, 0], [0, 1], [0, i], [0, j], [0, k], [0, E], [0, I], [0, J], [0, K]] ,
[[i,0], [- 1, 0], [k, 0], [- j, 0], [I, 0], [- E, 0], [- K, 0], [J, 0], [0, i], [0, - 1], [0, - k], [0, j], [0, - I], [0, E], [0, K], [0, - J]] ,
[[j,0], [- k, 0], [- 1, 0], [i, 0], [J, 0], [K, 0], [- E, 0], [- I, 0], [0, j], [0, k], [0, - 1], [0, - i], [0, - J], [0, - K], [0, E], [0, I]] ,
[[k,0], [j, 0], [- i, 0], [- 1, 0], [K, 0], [- J, 0], [I, 0], [- E, 0], [0, k], [0, - j], [0, i], [0, - 1], [0, - K], [0, J], [0, - I], [0, E]] ,
[[E,0], [- I, 0], [- J, 0], [- K, 0], [- 1, 0], [i, 0], [j, 0], [k, 0], [0, E], [0, I], [0, J], [0, K], [0, - 1], [0, - i], [0, - j], [0, - k]] ,
[[I,0], [E, 0], [- K, 0], [J, 0], [- i, 0], [- 1, 0], [- k, 0], [j, 0], [0, I], [0, - E], [0, K], [0, - J], [0, i], [0, - 1], [0, k], [0, - j]] ,
[[J,0], [K, 0], [E, 0], [- I, 0], [- j, 0], [k, 0], [- 1, 0], [- i, 0], [0, J], [0, - K], [0, - E], [0, I], [0, j], [0, - k], [0, - 1], [0, i]] ,
[[K,0], [- J, 0], [I, 0], [E, 0], [- k, 0], [- j, 0], [i, 0], [- 1, 0], [0, K], [0, J], [0, - I], [0, - E], [0, k], [0, j], [0, - i], [0, - 1]] ,
[[0,1], [0, - i], [0, - j], [0, - k], [0, - E], [0, - I], [0, - J], [0, - K], [- 1, 0], [i, 0], [j, 0], [k, 0], [E, 0], [I, 0], [J, 0], [K, 0]] ,
[[0,i], [0, 1], [0, - k], [0, j], [0, - I], [0, E], [0, K], [0, - J], [- i, 0], [- 1, 0], [- k, 0], [j, 0], [- I, 0], [E, 0], [K, 0], [- J, 0]] ,
[[0,j], [0, k], [0, 1], [0, - i], [0, - J], [0, - K], [0, E], [0, I], [- j, 0], [k, 0], [- 1, 0], [- i, 0], [- J, 0], [- K, 0], [E, 0], [I, 0]] ,
[[0,k], [0, - j], [0, i], [0, 1], [0, - K], [0, J], [0, - I], [0, E], [- k, 0], [- j, 0], [i, 0], [- 1, 0], [- K, 0], [J, 0], [- I, 0], [E, 0]] ,
[[0,E], [0, I], [0, J], [0, K], [0, 1], [0, - i], [0, - j], [0, - k], [- E, 0], [I, 0], [J, 0], [K, 0], [- 1, 0], [- i, 0], [- j, 0], [- k, 0]] ,
[[0,I], [0, - E], [0, K], [0, - J], [0, i], [0, 1], [0, k], [0, - j], [- I, 0], [- E, 0], [K, 0], [- J, 0], [i, 0], [- 1, 0], [k, 0], [- j, 0]] ,
[[0,J], [0, - K], [0, - E], [0, I], [0, j], [0, - k], [0, 1], [0, i], [- J, 0], [- K, 0], [- E, 0], [I, 0], [j, 0], [- k, 0], [- 1, 0], [i, 0]] ,
[[0,K], [0, J], [0, - I], [0, - E], [0, k], [0, j], [0, - i], [0, 1], [- K, 0], [J, 0], [- I, 0], [- E, 0], [k, 0], [j, 0], [- i, 0], [- 1, 0]] ]
Type: Matrix(DirectProduct?(2,
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caleyConj(x:SED):SED == pair(conjugate(x.1),-x.2)
Function declaration caleyConj : DirectProduct(2,Octonion(Expression (Integer))) -> DirectProduct(2, Octonion(Expression(Integer))) has been added to workspace.
Type: Void
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caleyInv(x:SED):SED == inv(caleyMul(caleyConj x,x).1) * caleyConj(x)
Function declaration caleyInv : DirectProduct(2,Octonion(Expression( Integer))) -> DirectProduct(2, Octonion(Expression(Integer))) has been added to workspace.
Type: Void
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--S0(y)==map(x+->(x*inv(y)=1 or x*inv(y)=-1 => x*inv(y);0),M0) --S0(B0.1) S(y)==map(x+->(caleyMul(x, caleyInv y)=caleyOne => 1;caleyMul(x, caleyInv y)=-caleyOne => -1;0), M)
Type: Void
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S(B.1)
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Compiling function caleyConj with type DirectProduct(2,Octonion( Expression(Integer))) -> DirectProduct(2, Octonion(Expression( Integer)))
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Compiling function caleyInv with type DirectProduct(2,Octonion( Expression(Integer))) -> DirectProduct(2, Octonion(Expression( Integer)))
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Compiling function S with type DirectProduct(2,Octonion(Expression( Integer))) -> Matrix(Integer)
(17) [[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, - 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, - 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, - 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]]
Type: Matrix(Integer)
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--Yg0:T:=unravel concat concat(map(S0,B0)::List List List R); Yg:T:=unravel concat concat(map(S, B)::List List List R)
(18) [matrix1,matrix2, matrix3, matrix4, matrix5, matrix6, matrix7, matrix8, matrix9, matrix10, matrix11, matrix12, matrix13, matrix14, matrix15, matrix16]
where matrix1 = [[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, - 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, - 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, - 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 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 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 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 - 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| and matrix2 = | | |0 0 0 0 0 0 0 0 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 - 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 - 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 1| | | +0 0 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 0 0 0 + | | |0 0 0 - 1 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 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 - 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 | and matrix3 = | | |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 - 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 - 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 1 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 0 0 + | | |0 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 | | | |1 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 - 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 - 1 0 0 0 0 0 0 0 0 0 0 0 | and matrix4 = | | |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 - 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 - 1 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 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 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 - 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| | | |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| and matrix5 = | | |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 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 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 - 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 - 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 - 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| and matrix6 = | | |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 - 1 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 - 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 - 1 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 - 1 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 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 - 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 | | | |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 | and matrix7 = | | |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 - 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 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 - 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 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 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 - 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 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| and matrix8 = | | |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 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 - 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 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 - 1 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 - 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 - 1| and matrix9 = | | |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 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 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 - 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 - 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 1| | | |0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 0| and matrix10 = | | |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 - 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 - 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 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 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 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 - 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 1 0 0 0 | | | |0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 | and matrix11 = | | |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 | | | |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 - 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 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 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 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 - 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 - 1 0 0 | | | |0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 | and matrix12 = | | |0 0 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 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 - 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 - 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 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 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| and matrix13 = | | |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| | | |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 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 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 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 1 0 0 0 0| | | |0 0 0 0 0 0 0 0 0 0 - 1 0 0 0 0 0| and matrix14 = | | |0 0 0 0 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 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 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 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 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 - 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 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 1 0 0 0 0 0 0 0 | | | |0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 | and matrix15 = | | |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 - 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 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 | | | |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 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 - 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 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 - 1 0 0 0 0 0 0| | | |0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0| and matrix16 = | | |0 0 0 0 0 0 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 - 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 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 - 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+
fricas
test(Yg0=Yg)
(19) false
Type: Boolean
A scalar product is denoted by the (2,0)-tensor
fricas
U:T := unravel(concat [[script(u,[[], [j, i]]) for i in 1..dim] for j in 1..dim] );
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
 | (1) |
fricas
ω := reindex(reindex(U,[2, 1])*reindex(Yg, [1, 3, 2]), [3, 2, 1])-U*Yg;
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.
fricas
)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))
fricas
uu := transpose matrix [concat(map(variables,ravel(U)))::List Symbol];
Type: Matrix(Polynomial(Integer))
fricas
J::OutputForm * uu::OutputForm = 0;
Type: Equation(OutputForm?)
fricas
nrows(J)
(25) 4096
Type: PositiveInteger?
fricas
ncols(J)
(26) 256
Type: PositiveInteger?
The matrix J transforms the coefficients of the tensor 
into coefficients of the tensor 
. We are looking for
the general linear family of tensors 
such that
J transforms into 
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:
fricas
NJ:=nullSpace(J)
(27) [ [- 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, 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, 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, 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] ]
Type: List(Vector(Expression(Integer)))
fricas
SS:=map((x,y)+->x=y, concat map(variables, ravel U), entries reduce(+, [p[i]*NJ.i for i in 1..#NJ]))
(28) 1,1 1, 2 1, 3 1, 4 1, 5 1, 6 [u = - p , u = 0, u = 0, u = 0, u = 0, u = 0, 1 1, 7 1, 8 1, 9 1, 10 1, 11 1, 12 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 1, 13 1, 14 1, 15 1, 16 2, 1 2, 2 u = 0, u = 0, u = 0, u = 0, u = 0, u = p , 1 2, 3 2, 4 2, 5 2, 6 2, 7 2, 8 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 2, 9 2, 10 2, 11 2, 12 2, 13 2, 14 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 2, 15 2, 16 3, 1 3, 2 3, 3 3, 4 u = 0, u = 0, u = 0, u = 0, u = p , u = 0, 1 3, 5 3, 6 3, 7 3, 8 3, 9 3, 10 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 3, 11 3, 12 3, 13 3, 14 3, 15 3, 16 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 4, 1 4, 2 4, 3 4, 4 4, 5 4, 6 u = 0, u = 0, u = 0, u = p , u = 0, u = 0, 1 4, 7 4, 8 4, 9 4, 10 4, 11 4, 12 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 4, 13 4, 14 4, 15 4, 16 5, 1 5, 2 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 5, 3 5, 4 5, 5 5, 6 5, 7 5, 8 u = 0, u = 0, u = p , u = 0, u = 0, u = 0, 1 5, 9 5, 10 5, 11 5, 12 5, 13 5, 14 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 5, 15 5, 16 6, 1 6, 2 6, 3 6, 4 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 6, 5 6, 6 6, 7 6, 8 6, 9 6, 10 u = 0, u = p , u = 0, u = 0, u = 0, u = 0, 1 6, 11 6, 12 6, 13 6, 14 6, 15 6, 16 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 7, 1 7, 2 7, 3 7, 4 7, 5 7, 6 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 7, 7 7, 8 7, 9 7, 10 7, 11 7, 12 u = p , u = 0, u = 0, u = 0, u = 0, u = 0, 1 7, 13 7, 14 7, 15 7, 16 8, 1 8, 2 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 8, 3 8, 4 8, 5 8, 6 8, 7 8, 8 u = 0, u = 0, u = 0, u = 0, u = 0, u = p , 1 8, 9 8, 10 8, 11 8, 12 8, 13 8, 14 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 8, 15 8, 16 9, 1 9, 2 9, 3 9, 4 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 9, 5 9, 6 9, 7 9, 8 9, 9 9, 10 u = 0, u = 0, u = 0, u = 0, u = p , u = 0, 1 9, 11 9, 12 9, 13 9, 14 9, 15 9, 16 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 10, 1 10, 2 10, 3 10, 4 10, 5 10, 6 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 10, 7 10, 8 10, 9 10, 10 10, 11 10, 12 u = 0, u = 0, u = 0, u = p , u = 0, u = 0, 1 10, 13 10, 14 10, 15 10, 16 11, 1 11, 2 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 11, 3 11, 4 11, 5 11, 6 11, 7 11, 8 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 11, 9 11, 10 11, 11 11, 12 11, 13 u = 0, u = 0, u = p , u = 0, u = 0, 1 11, 14 11, 15 11, 16 12, 1 12, 2 12, 3 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 12, 4 12, 5 12, 6 12, 7 12, 8 12, 9 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 12, 10 12, 11 12, 12 12, 13 12, 14 u = 0, u = 0, u = p , u = 0, u = 0, 1 12, 15 12, 16 13, 1 13, 2 13, 3 13, 4 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 13, 5 13, 6 13, 7 13, 8 13, 9 13, 10 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 13, 11 13, 12 13, 13 13, 14 13, 15 u = 0, u = 0, u = p , u = 0, u = 0, 1 13, 16 14, 1 14, 2 14, 3 14, 4 14, 5 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 14, 6 14, 7 14, 8 14, 9 14, 10 14, 11 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 14, 12 14, 13 14, 14 14, 15 14, 16 u = 0, u = 0, u = p , u = 0, u = 0, 1 15, 1 15, 2 15, 3 15, 4 15, 5 15, 6 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 15, 7 15, 8 15, 9 15, 10 15, 11 15, 12 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 15, 13 15, 14 15, 15 15, 16 16, 1 u = 0, u = 0, u = p , u = 0, u = 0, 1 16, 2 16, 3 16, 4 16, 5 16, 6 16, 7 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 16, 8 16, 9 16, 10 16, 11 16, 12 16, 13 u = 0, u = 0, u = 0, u = 0, u = 0, u = 0, 16, 14 16, 15 16, 16 u = 0, u = 0, u = p ] 1
Type: List(Equation(Expression(Integer)))
fricas
Ug:T := unravel(map(x+->subst(x,SS), ravel U))
+- p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + | 1 | | | | 0 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 p 0 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 p 0 0 0 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 p 0 0 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 p 0 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 p 0 0 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 p 0 0 0 0 0 0 0 0 | | 1 | (29) | | | 0 0 0 0 0 0 0 0 p 0 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 p 0 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 p 0 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 0 p 0 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 0 0 p 0 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 0 0 0 p 0 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p 0 | | 1 | | | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p | + 1+
This defines a family of pre-Frobenius algebras:
fricas
test(unravel(map(x+->subst(x,SS), ravel ω))$T=0*ω)
(30) true
Type: Boolean
The scalar product must be non-degenerate:
fricas
Ud:DMP([p[i] for i in 1..#NJ],INT) := determinant [[Ug[i, j] for j in 1..dim] for i in 1..dim]
16 (31) - p 1
Type: DistributedMultivariatePolynomial?([p[1]],
fricas
factor Ud
16 (32) - p 1
Type: Factored(DistributedMultivariatePolynomial?([p[1]],
Definition 3
Co-pairing
fricas
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)
+ 1 + |- -- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 | | p | | 1 | (33) | | | 1 | | 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 | | p | | 1 | | | | 1| | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 --| | p | + 1+
dimension Ω U
fricas
contract(contract(Ωg,1, Ug, 1), 1, 2)
(34) 16
Definition 4
Co-multiplication
fricas
λg:=reindex(contract(contract(Ug*Yg,1, Ωg, 1), 1, Ωg, 1), [2, 3, 1]);
fricas
-- just for display reindex(λg,[3, 1, 2])
(36) [matrix1,matrix2, matrix3, matrix4, matrix5, matrix6, matrix7, matrix8, matrix9, matrix10, matrix11, matrix12, matrix13, matrix14, matrix15, matrix16]
where matrix1 = + 1 + |- -- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 0 | | p | | 1 | | | | 1 | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -- 0 | | p | | 1 | | | | 1| | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 --| | p | + 1+
and matrix2 = 1 [[0,- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0]] p 1
and matrix3 = 1 [[0,0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0]] p 1
and matrix4 = 1 [[0,0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0]] p 1
and matrix5 = 1 [[0,0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0]] p 1
and matrix6 = 1 [[0,0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0]] p 1
and matrix7 = 1 [[0,0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0]] p 1
and matrix8 = 1 [[0,0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix9 = 1 [[0,0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix10 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix11 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix12 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix13 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix14 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix15 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
and matrix16 = 1 [[0,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, 0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, 0, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [0, - --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], p 1 1 [- --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]] p 1
i λ=Ω
fricas
test(λg*X(1)=Ωg)
(37) true
Type: Boolean
Definition 5
i U
fricas
ιg:=X(1)*Ug
(38) [- p ,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] 1
Y=U ι
fricas
test(ιg * Yg = Ug)
(39) true
Type: Boolean
For example:
fricas
Ug0:T:=unravel eval(ravel Ug,[p[1]=1])
+- 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 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| (40) | | | 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 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+
fricas
Ωg0:T:=unravel eval(ravel Ωg,[p[1]=1])
+- 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 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| (41) | | | 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 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+
fricas
λg0:T:=unravel eval(ravel λg,[p[1]=1]);
fricas
reindex(λg0,[3, 1, 2])
(43) +- 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 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 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 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 - 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 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 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 - 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 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 - 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 - 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 1| | | + 0 0 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 0 0 0 + | | | 0 0 0 - 1 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 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 - 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 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 - 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 - 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 1 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 0 0 + | | | 0 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 | | | |- 1 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 - 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 - 1 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 - 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 - 1 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 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 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 - 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| | | |- 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 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 - 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 - 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 - 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 - 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 - 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 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 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 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 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 - 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 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 - 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 | | | |- 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 1 0 | | | | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 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