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Grassmann Algebra Is Frobenius In Many Ways

A 2^n-dimensional vector space represents Grassmann algebra with n generators

Linear operators over a 4-dimensional vector space representing Grassmann algebra with two generators.

Ref:

We need the Axiom LinearOperator library.

fricas
)library CARTEN ARITY CMONAL CPROP CLOP CALEY
CartesianTensor is now explicitly exposed in frame initial CartesianTensor will be automatically loaded when needed from /var/aw/var/LatexWiki/CARTEN.NRLIB/CARTEN Arity is now explicitly exposed in frame initial Arity will be automatically loaded when needed from /var/aw/var/LatexWiki/ARITY.NRLIB/ARITY ClosedMonoidal is now explicitly exposed in frame initial ClosedMonoidal will be automatically loaded when needed from /var/aw/var/LatexWiki/CMONAL.NRLIB/CMONAL ClosedProp is now explicitly exposed in frame initial ClosedProp will be automatically loaded when needed from /var/aw/var/LatexWiki/CPROP.NRLIB/CPROP ClosedLinearOperator is now explicitly exposed in frame initial ClosedLinearOperator will be automatically loaded when needed from /var/aw/var/LatexWiki/CLOP.NRLIB/CLOP CaleyDickson is now explicitly exposed in frame initial CaleyDickson will be automatically loaded when needed from /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY

Use the following macros for convenient notation

fricas
-- summation
macro Σ(x,i,n)==reduce(+,[x for i in n])
Type: Void
fricas
-- list
macro Ξ(f,i,n)==[f for i in n]
Type: Void
fricas
-- subscript and superscripts
macro sb == subscript
Type: Void
fricas
macro sp == superscript
Type: Void

𝐋 is the domain of 4-dimensional linear operators over the rational functions ℚ (Expression Integer), i.e. ratio of polynomials with integer coefficients.

fricas
dim:=4

\label{eq1}4(1)
fricas
macro ℒ == List
Type: Void
fricas
macro ℂ == CaleyDickson
Type: Void
fricas
macro ℚ == Expression Integer
Type: Void
fricas
𝐋 := ClosedLinearOperator(OVAR ['1,'i,'j,'k], ℚ)

\label{eq2}\hbox{\axiomType{ClosedLinearOperator}\ } (\hbox{\axiomType{OrderedVariableList}\ } ([ 1, i , j , k ]) , \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }))(2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()

\label{eq3}\left[{|_{\  1}}, \:{|_{\  i}}, \:{|_{\  j}}, \:{|_{\  k}}\right](3)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()

\label{eq4}\left[{|^{\  1}}, \:{|^{\  i}}, \:{|^{\  j}}, \:{|^{\  k}}\right](4)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition

\label{eq5}{|_{\  1}^{\  1}}+{|_{\  i}^{\  i}}+{|_{\  j}^{\  j}}+{|_{\  k}^{\  k}}(5)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist

\label{eq6}\begin{array}{@{}l}
\displaystyle
{|_{\  1 \  1}^{\  1 \  1}}+{|_{\  i \  1}^{\  1 \  i}}+{|_{\  j \  1}^{\  1 \  j}}+{|_{\  k \  1}^{\  1 \  k}}+{|_{\  1 \  i}^{\  i \  1}}+ 
\
\
\displaystyle
{|_{\  i \  i}^{\  i \  i}}+{|_{\  j \  i}^{\  i \  j}}+{|_{\  k \  i}^{\  i \  k}}+{|_{\  1 \  j}^{\  j \  1}}+{|_{\  i \  j}^{\  j \  i}}+{|_{\  j \  j}^{\  j \  j}}+ 
\
\
\displaystyle
{|_{\  k \  j}^{\  j \  k}}+{|_{\  1 \  k}^{\  k \  1}}+{|_{\  i \  k}^{\  k \  i}}+{|_{\  j \  k}^{\  k \  j}}+{|_{\  k \  k}^{\  k \  k}}
(6)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
V:𝐋:=ev(1) -- evaluation

\label{eq7}{|^{\  1 \  1}}+{|^{\  i \  i}}+{|^{\  j \  j}}+{|^{\  k \  k}}(7)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
Λ:𝐋:=co(1) -- co-evaluation

\label{eq8}{|_{\  1 \  1}}+{|_{\  i \  i}}+{|_{\  j \  j}}+{|_{\  k \  k}}(8)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
equate(eq)==map((x,y)+->(x=y),ravel lhs eq, ravel rhs eq);
Type: Void

Generate structure constants for Grassmann Algebra

The structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Grassmann algebra will be specified by setting the Caley-Dickson parameters (i2, j2) to zero.

fricas
i2:=sp('i,[2])

\label{eq9}i^{2}(9)
Type: Symbol
fricas
j2:=sp('j,[2])

\label{eq10}j^{2}(10)
Type: Symbol
fricas
QQ:=CliffordAlgebra(2,ℚ,matrix [[i2,0],[0,j2]])

\label{eq11}\hbox{\axiomType{CliffordAlgebra}\ } (2, \hbox{\axiomType{Expression}\ } (\hbox{\axiomType{Integer}\ }) , [ [ i [ ; 2 ] , 0 ] , [ 0, j [ ; 2 ] ] ])(11)
Type: Type
fricas
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[1,2])]

\label{eq12}\left[ 1, \:{e_{1}}, \:{e_{2}}, \:{{e_{1}}\ {e_{2}}}\right](12)
Type: List(CliffordAlgebra(2,Expression(Integer),[[i[;2],0],[0,j[;2]]]))
fricas
M:Matrix QQ := matrix Ξ(Ξ(B.i*B.j, i,1..dim), j,1..dim)

\label{eq13}\left[ 
\begin{array}{cccc}
1 &{e_{1}}&{e_{2}}&{{e_{1}}\ {e_{2}}}
\
{e_{1}}&{i^{2}}& -{{e_{1}}\ {e_{2}}}& -{{i^{2}}\ {e_{2}}}
\
{e_{2}}&{{e_{1}}\ {e_{2}}}&{j^{2}}&{{j^{2}}\ {e_{1}}}
\
{{e_{1}}\ {e_{2}}}&{{i^{2}}\ {e_{2}}}& -{{j^{2}}\ {e_{1}}}& -{{i^{2}}\ {j^{2}}}
(13)
Type: Matrix(CliffordAlgebra(2,Expression(Integer),[[i[;2],0],[0,j[;2]]]))
fricas
S(y) == map(x +-> coefficient(recip(y)*x,[]),M)
Type: Void
fricas
ѕ :=map(S,B)::ℒ ℒ ℒ ℚ
fricas
Compiling function S with type CliffordAlgebra(2,Expression(Integer)
      ,[[i[;2],0],[0,j[;2]]]) -> Matrix(Expression(Integer))

\label{eq14}\begin{array}{@{}l}
\displaystyle
\left[{\left[{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \:{j^{2}}, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: -{{i^{2}}\ {j^{2}}}\right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 1, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \:{j^{2}}\right]}, \:{\left[ 0, \: 0, \: -{j^{2}}, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 1, \: 0 \right]}, \:{\left[ 0, \: 0, \: 0, \: -{i^{2}}\right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}, \:{\left[ 0, \:{i^{2}}, \: 0, \: 0 \right]}\right]}, \: \right.
\
\
\displaystyle
\left.{\left[{\left[ 0, \: 0, \: 0, \: 1 \right]}, \:{\left[ 0, \: 0, \: - 1, \: 0 \right]}, \:{\left[ 0, \: 1, \: 0, \: 0 \right]}, \:{\left[ 1, \: 0, \: 0, \: 0 \right]}\right]}\right] (14)
Type: List(List(List(Expression(Integer))))
fricas
-- structure constants form a tensor operator
--Y := Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim)
Y := eval(Σ(Σ(Σ(ѕ(i)(k)(j)*𝐞.i*𝐝.j*𝐝.k, i,1..dim), j,1..dim), k,1..dim),[i2=0,j2=0])

\label{eq15}\begin{array}{@{}l}
\displaystyle
{|_{\  1}^{\  1 \  1}}+{|_{\  i}^{\  1 \  i}}+{|_{\  j}^{\  1 \  j}}+{|_{\  k}^{\  1 \  k}}+{|_{\  i}^{\  i \  1}}+{|_{\  k}^{\  i \  j}}+{|_{\  j}^{\  j \  1}}- 
\
\
\displaystyle
{|_{\  k}^{\  j \  i}}+{|_{\  k}^{\  k \  1}}
(15)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
matrix Ξ(Ξ((𝐞.i*𝐞.j)/Y, i,1..dim), j,1..dim)

\label{eq16}\left[ 
\begin{array}{cccc}
{|_{\  1}}&{|_{\  i}}&{|_{\  j}}&{|_{\  k}}
\
{|_{\  i}}& 0 & -{|_{\  k}}& 0 
\
{|_{\  j}}&{|_{\  k}}& 0 & 0 
\
{|_{\  k}}& 0 & 0 & 0 
(16)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))

Units

fricas
e:=𝐞.1; i:=𝐞.2; j:=𝐞.3; k:=𝐞.4;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Multiplication of arbitrary Grassmann numbers a and b

fricas
a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)

\label{eq17}{{a_{1}}\ {|_{\  1}}}+{{a_{2}}\ {|_{\  i}}}+{{a_{3}}\ {|_{\  j}}}+{{a_{4}}\ {|_{\  k}}}(17)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)

\label{eq18}{{b_{1}}\ {|_{\  1}}}+{{b_{2}}\ {|_{\  i}}}+{{b_{3}}\ {|_{\  j}}}+{{b_{4}}\ {|_{\  k}}}(18)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
(a*b)/Y

\label{eq19}\begin{array}{@{}l}
\displaystyle
{{a_{1}}\ {b_{1}}\ {|_{\  1}}}+{{\left({{a_{1}}\ {b_{2}}}+{{a_{2}}\ {b_{1}}}\right)}\ {|_{\  i}}}+{{\left({{a_{1}}\ {b_{3}}}+{{a_{3}}\ {b_{1}}}\right)}\ {|_{\  j}}}+ 
\
\
\displaystyle
{{\left({{a_{1}}\ {b_{4}}}+{{a_{2}}\ {b_{3}}}-{{a_{3}}\ {b_{2}}}+{{a_{4}}\ {b_{1}}}\right)}\ {|_{\  k}}}
(19)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )

\label{eq20} \mbox{\rm true} (20)
Type: Boolean

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

fricas
U:=Σ(Σ(script('u,[[],[i,j]])*𝐝.i*𝐝.j, i,1..dim), j,1..dim)

\label{eq21}\begin{array}{@{}l}
\displaystyle
{{u^{1, \: 1}}\ {|^{\  1 \  1}}}+{{u^{1, \: 2}}\ {|^{\  1 \  i}}}+{{u^{1, \: 3}}\ {|^{\  1 \  j}}}+{{u^{1, \: 4}}\ {|^{\  1 \  k}}}+ 
\
\
\displaystyle
{{u^{2, \: 1}}\ {|^{\  i \  1}}}+{{u^{2, \: 2}}\ {|^{\  i \  i}}}+{{u^{2, \: 3}}\ {|^{\  i \  j}}}+{{u^{2, \: 4}}\ {|^{\  i \  k}}}+{{u^{3, \: 1}}\ {|^{\  j \  1}}}+ 
\
\
\displaystyle
{{u^{3, \: 2}}\ {|^{\  j \  i}}}+{{u^{3, \: 3}}\ {|^{\  j \  j}}}+{{u^{3, \: 4}}\ {|^{\  j \  k}}}+{{u^{4, \: 1}}\ {|^{\  k \  1}}}+ 
\
\
\displaystyle
{{u^{4, \: 2}}\ {|^{\  k \  i}}}+{{u^{4, \: 3}}\ {|^{\  k \  j}}}+{{u^{4, \: 4}}\ {|^{\  k \  k}}}
(21)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),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:


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\label{eq22}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (22)
(three-point function) is zero.

Using the LinearOperator domain in Axiom and some carefully chosen symbols we can easily enter expressions that are both readable and interpreted by Axiom as "graphical calculus" diagrams describing complex products and compositions of linear operators.

fricas
ω:𝐋 :=
     (    Y I    )  /
           U        -
     (    I Y    )  /
           U

\label{eq23}\begin{array}{@{}l}
\displaystyle
{{\left({u^{2, \: 1}}-{u^{1, \: 2}}\right)}\ {|^{\  1 \  i \  1}}}+{{u^{2, \: 2}}\ {|^{\  1 \  i \  i}}}+{{\left({u^{2, \: 3}}-{u^{1, \: 4}}\right)}\ {|^{\  1 \  i \  j}}}+ 
\
\
\displaystyle
{{u^{2, \: 4}}\ {|^{\  1 \  i \  k}}}+{{\left({u^{3, \: 1}}-{u^{1, \: 3}}\right)}\ {|^{\  1 \  j \  1}}}+{{\left({u^{3, \: 2}}+{u^{1, \: 4}}\right)}\ {|^{\  1 \  j \  i}}}+ 
\
\
\displaystyle
{{u^{3, \: 3}}\ {|^{\  1 \  j \  j}}}+{{u^{3, \: 4}}\ {|^{\  1 \  j \  k}}}+{{\left({u^{4, \: 1}}-{u^{1, \: 4}}\right)}\ {|^{\  1 \  k \  1}}}+ 
\
\
\displaystyle
{{u^{4, \: 2}}\ {|^{\  1 \  k \  i}}}+{{u^{4, \: 3}}\ {|^{\  1 \  k \  j}}}+{{u^{4, \: 4}}\ {|^{\  1 \  k \  k}}}-{{u^{2, \: 2}}\ {|^{\  i \  i \  1}}}- 
\
\
\displaystyle
{{u^{2, \: 4}}\ {|^{\  i \  i \  j}}}+{{\left({u^{4, \: 1}}-{u^{2, \: 3}}\right)}\ {|^{\  i \  j \  1}}}+{{\left({u^{4, \: 2}}+{u^{2, \: 4}}\right)}\ {|^{\  i \  j \  i}}}+ 
\
\
\displaystyle
{{u^{4, \: 3}}\ {|^{\  i \  j \  j}}}+{{u^{4, \: 4}}\ {|^{\  i \  j \  k}}}-{{u^{2, \: 4}}\ {|^{\  i \  k \  1}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 1}}-{u^{3, \: 2}}\right)}\ {|^{\  j \  i \  1}}}-{{u^{4, \: 2}}\ {|^{\  j \  i \  i}}}+ 
\
\
\displaystyle
{{\left(-{u^{4, \: 3}}-{u^{3, \: 4}}\right)}\ {|^{\  j \  i \  j}}}-{{u^{4, \: 4}}\ {|^{\  j \  i \  k}}}-{{u^{3, \: 3}}\ {|^{\  j \  j \  1}}}+ 
\
\
\displaystyle
{{u^{3, \: 4}}\ {|^{\  j \  j \  i}}}-{{u^{3, \: 4}}\ {|^{\  j \  k \  1}}}-{{u^{4, \: 2}}\ {|^{\  k \  i \  1}}}-{{u^{4, \: 4}}\ {|^{\  k \  i \  j}}}- 
\
\
\displaystyle
{{u^{4, \: 3}}\ {|^{\  k \  j \  1}}}+{{u^{4, \: 4}}\ {|^{\  k \  j \  i}}}-{{u^{4, \: 4}}\ {|^{\  k \  k \  1}}}
(23)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Definition 2

An algebra with a non-degenerate associative scalar product is called a [Frobenius Algebra]?.

The Cartan-Killing Trace

fricas
Ú:=
   (  Y Λ  ) / _
   (   Y I ) / _
        V

\label{eq24}4 \ {|^{\  1 \  1}}(24)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
Ù:=
   (  Λ Y  ) / _
   ( I Y   ) / _
      V

\label{eq25}4 \ {|^{\  1 \  1}}(25)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(Ù=Ú)

\label{eq26} \mbox{\rm true} (26)
Type: Boolean

forms is degenerate

fricas
Ũ := Ù

\label{eq27}4 \ {|^{\  1 \  1}}(27)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ

\label{eq28} \mbox{\rm true} (28)
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)

\label{eq29}0(29)
Type: Expression(Integer)

General Solution

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.

fricas
)expose MCALCFN
MultiVariableCalculusFunctions is now explicitly exposed in frame initial J := jacobian(ravel ω,concat map(variables,ravel U)::ℒ Symbol);
Type: Matrix(Expression(Integer))
fricas
nrows(J),ncols(J)

\label{eq30}\left[{64}, \:{16}\right](30)
Type: Tuple(PositiveInteger)

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:

fricas
Ñ:=nullSpace(J)

\label{eq31}\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] (31)
Type: List(Vector(Expression(Integer)))
fricas
ℰ:=map((x,y)+->x=y, concat
       map(variables,ravel U), entries Σ(sb('p,[i])*Ñ.i, i,1..#Ñ) )

\label{eq32}\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] 
(32)
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

fricas
zero? eval(ω,ℰ)

\label{eq33} \mbox{\rm true} (33)
Type: Boolean
fricas
Ų:𝐋 := eval(U,ℰ)

\label{eq34}\begin{array}{@{}l}
\displaystyle
{{p_{1}}\ {|^{\  1 \  1}}}+{{p_{2}}\ {|^{\  1 \  i}}}+{{p_{3}}\ {|^{\  1 \  j}}}+{{p_{4}}\ {|^{\  1 \  k}}}+{{p_{2}}\ {|^{\  i \  1}}}+{{p_{4}}\ {|^{\  i \  j}}}+ 
\
\
\displaystyle
{{p_{3}}\ {|^{\  j \  1}}}-{{p_{4}}\ {|^{\  j \  i}}}+{{p_{4}}\ {|^{\  k \  1}}}
(34)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Frobenius Form (co-unit)

fricas
d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4

\label{eq35}{�� 1 \ {|^{\  1}}}+{�� i \ {|^{\  i}}}+{�� j \ {|^{\  j}}}+{�� k \ {|^{\  k}}}(35)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
𝔇:=equate(d=
    (    e I   ) / _
          Ų    )
fricas
Compiling function equate with type Equation(ClosedLinearOperator(
      OrderedVariableList([1,i,j,k]),Expression(Integer))) -> List(
      Equation(Expression(Integer)))

\label{eq36}\left[{�� 1 ={p_{1}}}, \:{�� i ={p_{2}}}, \:{�� j ={p_{3}}}, \:{�� k ={p_{4}}}\right](36)
Type: List(Equation(Expression(Integer)))

Express scalar product in terms of Frobenius form

fricas
𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1

\label{eq37}\left[{{p_{1}}= �� 1}, \:{{p_{2}}= �� i}, \:{{p_{3}}= �� j}, \:{{p_{4}}= �� k}\right](37)
Type: List(Equation(Expression(Integer)))
fricas
Ų:=eval(Ų,𝔓)

\label{eq38}\begin{array}{@{}l}
\displaystyle
{�� 1 \ {|^{\  1 \  1}}}+{�� i \ {|^{\  1 \  i}}}+{�� j \ {|^{\  1 \  j}}}+{�� k \ {|^{\  1 \  k}}}+{�� i \ {|^{\  i \  1}}}+{�� k \ {|^{\  i \  j}}}+{�� j \ {|^{\  j \  1}}}- 
\
\
\displaystyle
{�� k \ {|^{\  j \  i}}}+{�� k \ {|^{\  k \  1}}}
(38)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
        Y     /
        d     =  Ų

\label{eq39} \mbox{\rm true} (39)
Type: Boolean

In general the pairing is not symmetric!

fricas
u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)

\label{eq40}\left[ 
\begin{array}{cccc}
�� 1 & �� i & �� j & �� k 
\
�� i & 0 & - �� k & 0 
\
�� j & �� k & 0 & 0 
\
�� k & 0 & 0 & 0 
(40)
Type: Matrix(Expression(Integer))

fricas
)set output algebra on
 
fricas
)set output tex off

fricas
eigenvectors(u1::Matrix FRAC POLY INT)
(51) [ 4 2 2 2 2 2 3 4 [eigval = (%B | - εk - %B ε1 εk - %B εj - %B εi - %B ε1 + %B ), eigmult = 1,
eigvec = [ %B [[--], εk
[ 3 2 2 3 - εi εk - %B εj εk + (- %B ε1 + %B )εi εk - %B εj + 2 2 3 (- %B εi - %B ε1 + %B )εj / 2 2 2 (εj + εi )εk ] ,
[ 3 2 2 2 3 - εj εk + %B εi εk + (- %B ε1 + %B )εj εk + %B εi εj + %B εi + 2 3 (%B ε1 - %B )εi / 2 2 2 (εj + εi )εk ] , [1]] ] ] ]
Type: List(Record(eigval: Union(Fraction(Polynomial(Integer)),SuchThat?(Symbol,Polynomial(Integer))),eigmult: NonNegativeInteger,eigvec: List(Matrix(Fraction(Polynomial(Integer))))))

fricas
)set output algebra off
 
fricas
)set output tex on

The scalar product must be non-degenerate:

fricas
Ů:=determinant u1

\label{eq41}-{{�� k}^{4}}(41)
Type: Expression(Integer)
fricas
factor(numer Ů)/factor(denom Ů)

\label{eq42}-{{�� k}^{4}}(42)
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Frobenius scalar product of "vectors" a and b

fricas
a:=sb('a,[1])*i+sb('a,[2])*j

\label{eq43}{{a_{1}}\ {|_{\  i}}}+{{a_{2}}\ {|_{\  j}}}(43)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
b:=sb('b,[1])*i+sb('b,[2])*j

\label{eq44}{{b_{1}}\ {|_{\  i}}}+{{b_{2}}\ {|_{\  j}}}(44)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
(a,a)/Ų

\label{eq45}0(45)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
(a,b)/Ų

\label{eq46}{\left({{a_{1}}\ {b_{2}}}-{{a_{2}}\ {b_{1}}}\right)}\  �� k(46)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Definition 3

Co-scalar product

Solve the Snake Relation as a system of linear equations.

fricas
Ω:𝐋:=Σ(Σ(script('u,[[i,j]])*𝐞.i*𝐞.j, i,1..dim), j,1..dim)

\label{eq47}\begin{array}{@{}l}
\displaystyle
{{u_{1, \: 1}}\ {|_{\  1 \  1}}}+{{u_{1, \: 2}}\ {|_{\  1 \  i}}}+{{u_{1, \: 3}}\ {|_{\  1 \  j}}}+{{u_{1, \: 4}}\ {|_{\  1 \  k}}}+ 
\
\
\displaystyle
{{u_{2, \: 1}}\ {|_{\  i \  1}}}+{{u_{2, \: 2}}\ {|_{\  i \  i}}}+{{u_{2, \: 3}}\ {|_{\  i \  j}}}+{{u_{2, \: 4}}\ {|_{\  i \  k}}}+{{u_{3, \: 1}}\ {|_{\  j \  1}}}+ 
\
\
\displaystyle
{{u_{3, \: 2}}\ {|_{\  j \  i}}}+{{u_{3, \: 3}}\ {|_{\  j \  j}}}+{{u_{3, \: 4}}\ {|_{\  j \  k}}}+{{u_{4, \: 1}}\ {|_{\  k \  1}}}+ 
\
\
\displaystyle
{{u_{4, \: 2}}\ {|_{\  k \  i}}}+{{u_{4, \: 3}}\ {|_{\  k \  j}}}+{{u_{4, \: 4}}\ {|_{\  k \  k}}}
(47)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
UXΩ:=(I*ΩX)/(Ų*I);
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
ΩXU:=(ΩX*I)/(I*Ų);
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
eq1:=equate(UXΩ=I);
Type: List(Equation(Expression(Integer)))
fricas
eq2:=equate(ΩXU=I);
Type: List(Equation(Expression(Integer)))
fricas
snake:=solve(concat(eq1,eq2),concat Ξ(Ξ(script('u,[[i,j]]), i,1..dim), j,1..dim));
Type: List(List(Equation(Expression(Integer))))
fricas
if #snake ~= 1 then error "no solution"
Type: Void
fricas
Ω:=eval(Ω,snake(1))

\label{eq48}\begin{array}{@{}l}
\displaystyle
{{1 \over �� k}\ {|_{\  1 \  k}}}+{{1 \over �� k}\ {|_{\  i \  j}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  i \  k}}}-{{1 \over �� k}\ {|_{\  j \  i}}}+{{�� i \over{{�� k}^{2}}}\ {|_{\  j \  k}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  k \  1}}}+{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  i}}}-{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  j}}}-{{�� 1 \over{{�� k}^{2}}}\ {|_{\  k \  k}}}
(48)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
ΩX:=Ω/X;
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

fricas
matrix Ξ(Ξ(retract(Ω/(𝐝.i*𝐝.j)), i,1..dim), j,1..dim)

\label{eq49}\left[ 
\begin{array}{cccc}
0 & 0 & 0 &{1 \over �� k}
\
0 & 0 & -{1 \over �� k}&{�� j \over{{�� k}^{2}}}
\
0 &{1 \over �� k}& 0 & -{�� i \over{{�� k}^{2}}}
\
{1 \over �� k}& -{�� j \over{{�� k}^{2}}}&{�� i \over{{�� k}^{2}}}& -{�� 1 \over{{�� k}^{2}}}
(49)
Type: Matrix(Expression(Integer))

Check "dimension" and the snake relations.

fricas
O:𝐋:=
       Ω    /
       Ų

\label{eq50}4(50)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ų I     )  =  I

\label{eq51} \mbox{\rm true} (51)
Type: Boolean
fricas
test
    (     ΩX I    )  /
    (    I Ų      )  =  I

\label{eq52} \mbox{\rm true} (52)
Type: Boolean

Definition 4

Co-algebra

Compute the "three-point" function and use it to define co-multiplication.

fricas
W:=
  (Y I) /
    Ų

\label{eq53}\begin{array}{@{}l}
\displaystyle
{�� 1 \ {|^{\  1 \  1 \  1}}}+{�� i \ {|^{\  1 \  1 \  i}}}+{�� j \ {|^{\  1 \  1 \  j}}}+{�� k \ {|^{\  1 \  1 \  k}}}+{�� i \ {|^{\  1 \  i \  1}}}+ 
\
\
\displaystyle
{�� k \ {|^{\  1 \  i \  j}}}+{�� j \ {|^{\  1 \  j \  1}}}-{�� k \ {|^{\  1 \  j \  i}}}+{�� k \ {|^{\  1 \  k \  1}}}+{�� i \ {|^{\  i \  1 \  1}}}+ 
\
\
\displaystyle
{�� k \ {|^{\  i \  1 \  j}}}+{�� k \ {|^{\  i \  j \  1}}}+{�� j \ {|^{\  j \  1 \  1}}}-{�� k \ {|^{\  j \  1 \  i}}}-{�� k \ {|^{\  j \  i \  1}}}+ 
\
\
\displaystyle
{�� k \ {|^{\  k \  1 \  1}}}
(53)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
λ:=
  (  ΩX I ΩX  ) /
  (  I  W  I  )

\label{eq54}\begin{array}{@{}l}
\displaystyle
{{1 \over �� k}\ {|_{\  1 \  k}^{\  1}}}-{{1 \over �� k}\ {|_{\  i \  j}^{\  1}}}+{{�� j \over{{�� k}^{2}}}\ {|_{\  i \  k}^{\  1}}}+{{1 \over �� k}\ {|_{\  j \  i}^{\  1}}}- 
\
\
\displaystyle
{{�� i \over{{�� k}^{2}}}\ {|_{\  j \  k}^{\  1}}}+{{1 \over �� k}\ {|_{\  k \  1}^{\  1}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  i}^{\  1}}}+{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  j}^{\  1}}}- 
\
\
\displaystyle
{{�� 1 \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1}}}+{{1 \over �� k}\ {|_{\  i \  k}^{\  i}}}+{{1 \over �� k}\ {|_{\  k \  i}^{\  i}}}-{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  i}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  j \  k}^{\  j}}}+{{1 \over �� k}\ {|_{\  k \  j}^{\  j}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  j}}}+{{1 \over �� k}\ {|_{\  k \  k}^{\  k}}}
(54)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

fricas
test
     (    I ΩX     )  /
     (     Y I     )  =  λ

\label{eq55} \mbox{\rm true} (55)
Type: Boolean
fricas
test
     (     ΩX I    )  /
     (    I  Y     )  =  λ

\label{eq56} \mbox{\rm true} (56)
Type: Boolean

Co-associativity

fricas
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )

\label{eq57} \mbox{\rm true} (57)
Type: Boolean

fricas
test
         e     /
         λ     =    ΩX

\label{eq58} \mbox{\rm true} (58)
Type: Boolean

Frobenius Condition (fork)

fricas
H :=
         Y    /
         λ

\label{eq59}\begin{array}{@{}l}
\displaystyle
{{1 \over �� k}\ {|_{\  1 \  k}^{\  1 \  1}}}-{{1 \over �� k}\ {|_{\  i \  j}^{\  1 \  1}}}+{{�� j \over{{�� k}^{2}}}\ {|_{\  i \  k}^{\  1 \  1}}}+{{1 \over �� k}\ {|_{\  j \  i}^{\  1 \  1}}}- 
\
\
\displaystyle
{{�� i \over{{�� k}^{2}}}\ {|_{\  j \  k}^{\  1 \  1}}}+{{1 \over �� k}\ {|_{\  k \  1}^{\  1 \  1}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  i}^{\  1 \  1}}}+{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  j}^{\  1 \  1}}}- 
\
\
\displaystyle
{{�� 1 \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1 \  1}}}+{{1 \over �� k}\ {|_{\  i \  k}^{\  1 \  i}}}+{{1 \over �� k}\ {|_{\  k \  i}^{\  1 \  i}}}-{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1 \  i}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  j \  k}^{\  1 \  j}}}+{{1 \over �� k}\ {|_{\  k \  j}^{\  1 \  j}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1 \  j}}}+{{1 \over �� k}\ {|_{\  k \  k}^{\  1 \  k}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  i \  k}^{\  i \  1}}}+{{1 \over �� k}\ {|_{\  k \  i}^{\  i \  1}}}-{{�� i \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  i \  1}}}+{{1 \over �� k}\ {|_{\  k \  k}^{\  i \  j}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  j \  k}^{\  j \  1}}}+{{1 \over �� k}\ {|_{\  k \  j}^{\  j \  1}}}-{{�� j \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  j \  1}}}-{{1 \over �� k}\ {|_{\  k \  k}^{\  j \  i}}}+ 
\
\
\displaystyle
{{1 \over �� k}\ {|_{\  k \  k}^{\  k \  1}}}
(59)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H

\label{eq60} \mbox{\rm true} (60)
Type: Boolean
fricas
test
     (   I λ   )  /
     (    Y I  )  =  H

\label{eq61} \mbox{\rm true} (61)
Type: Boolean

Handle

fricas
Φ :=
         λ     /
         Y

\label{eq62}0(62)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Figure 12

fricas
φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y )

\label{eq63}{4 \over{{�� k}^{2}}}\ {|_{\  k \  k}}(63)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y Y  )

\label{eq64}{4 \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1 \  1}}(64)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test((e,e)/ΦΦ=φφ)

\label{eq65} \mbox{\rm true} (65)
Type: Boolean

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I X I ) / _
  (  Y Y  )

\label{eq66}{4 \over{{�� k}^{2}}}\ {|_{\  k \  k}^{\  1 \  1}}(66)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test((e,e)/ΦΦ=φφ)

\label{eq67} \mbox{\rm true} (67)
Type: Boolean

Y-forms

Three traces of two graftings of an algebra gives six (2,0)-forms.

Left snail and right snail:

  LS                    RS

  Y /\                    /\ Y
   Y  )                  (  Y
    \/                    \/

  i  j                        j  i
   \/                          \/
    \    /\              /\    /
     e  f  \            /  f  e
      \/    \          /    \/
       \    /          \    /
        f  /            \  f
         \/              \/

fricas
LS:=
  ( Y Λ  )/ _
  (  Y I )/ _
      V

\label{eq68}4 \ {|^{\  1 \  1}}(68)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
RS:=
  (  Λ Y )/ _
  ( I Y  )/ _
     V

\label{eq69}4 \ {|^{\  1 \  1}}(69)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LS=RS)

\label{eq70} \mbox{\rm true} (70)
Type: Boolean

Left and right deer:

   RD                 LD

   \ /\/              \/\ /
    Y /\              /\ Y
     Y  )            (  Y
      \/              \/

   i            j    i            j
    \    /\    /      \    /\    /
     \  f  \  /        \  /  f  /
      \/    \/          \/    \/
       \    /\          /\    /
        e  /  \        /  \  e
         \/    \      /    \/
          \    /      \    /
           f  /        \  f
            \/          \/

Left and right deer forms are identical but different from snails.

fricas
RD:=
  (  I Λ I  ) / _
  (   Y X   ) / _
  (    Y I  ) / _
        V

\label{eq71}4 \ {|^{\  1 \  1}}(71)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
LD:=
  (  I Λ I  ) / _
  (   X Y   ) / _
  (  I Y    ) / _
      V

\label{eq72}4 \ {|^{\  1 \  1}}(72)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LD=RD)

\label{eq73} \mbox{\rm true} (73)
Type: Boolean
fricas
test(RD=RS)

\label{eq74} \mbox{\rm true} (74)
Type: Boolean
fricas
test(RD=LS)

\label{eq75} \mbox{\rm true} (75)
Type: Boolean

Left and right turtles:

  RT                   LT

   /\ / /               \ \ /\
  (  Y /                 \ Y  )
   \  Y                   Y  /
    \/                     \/

           i     j      i     j
    /\    /     /        \     \    /\
   /  f  /     /          \     \  f  \
  /    \/     /            \     \/    \
  \     \    /              \    /     /
   \     e  /                \  e     /
    \     \/                  \/     /
     \    /                    \    /
      \  f                      f  /
       \/                        \/

fricas
RT:=
  (  Λ I I ) / _
  ( I Y I  ) / _
  (  I Y   ) / _
      V

\label{eq76}4 \ {|^{\  1 \  1}}(76)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
LT:=
  ( I I Λ  ) / _
  (  I Y I ) / _
  (   Y I  ) / _
       V

\label{eq77}4 \ {|^{\  1 \  1}}(77)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LT=RT)

\label{eq78} \mbox{\rm true} (78)
Type: Boolean

The turles are symmetric

fricas
test(RT = X/RT)

\label{eq79} \mbox{\rm true} (79)
Type: Boolean
fricas
test(LT = X/LT)

\label{eq80} \mbox{\rm true} (80)
Type: Boolean

Five of the six forms are independent.

fricas
test(RT=RS)

\label{eq81} \mbox{\rm true} (81)
Type: Boolean
fricas
test(RT=LS)

\label{eq82} \mbox{\rm true} (82)
Type: Boolean
fricas
test(RT=RD)

\label{eq83} \mbox{\rm true} (83)
Type: Boolean
fricas
test(LT=RS)

\label{eq84} \mbox{\rm true} (84)
Type: Boolean
fricas
test(LT=LS)

\label{eq85} \mbox{\rm true} (85)
Type: Boolean
fricas
test(LT=RD)

\label{eq86} \mbox{\rm true} (86)
Type: Boolean




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