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Grassmann Algebra Is Frobenius In Many Ways

A -dimensional vector space represents Grassmann algebra with 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

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-- 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.

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dim:=4
 (1)
Type: PositiveInteger?
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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)
 (2)
Type: Type
fricas
𝐞:ℒ 𝐋      := basisOut()
 (3)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))
fricas
𝐝:ℒ 𝐋      := basisIn()
 (4)
Type: List(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))
fricas
I:𝐋:=[1]   -- identity for composition
 (5)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
X:𝐋:=[2,1] -- twist
 (6)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
V:𝐋:=ev(1) -- evaluation
 (7)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
Λ:𝐋:=co(1) -- co-evaluation
 (8)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
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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.

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i2:=sp('i,[2])
 (9)
Type: Symbol
fricas
j2:=sp('j,[2])
 (10)
Type: Symbol
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QQ:=CliffordAlgebra(2,ℚ,matrix [[i2,0],[0,j2]])
 (11)
Type: Type
fricas
B:ℒ QQ := [monomial(1,[]),monomial(1,[1]),monomial(1,[2]),monomial(1,[1,2])]
 (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)
 (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
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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))
 (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])
 (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)
 (16)
Type: Matrix(ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)))

Units

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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 and

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a:=Σ(sb('a,[i])*𝐞.i, i,1..dim)
 (17)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
b:=Σ(sb('b,[i])*𝐞.i, i,1..dim)
 (18)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
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(a*b)/Y
 (19)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Multiplication is Associative

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test(
( I Y ) / _
(  Y  ) = _
( Y I ) / _
(  Y  ) )
 (20)
Type: Boolean

A scalar product is denoted by the (2,0)-tensor

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

 (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
 (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
 (24)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
Ù:=
(  Λ Y  ) / _
( I Y   ) / _
V
 (25)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(Ù=Ú)
 (26)
Type: Boolean

forms is degenerate

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Ũ := Ù
 (27)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
(    Y I    )  /
Ũ        =
(    I Y    )  /
Ũ
 (28)
Type: Boolean
fricas
determinant Ξ(Ξ(retract((𝐞.i * 𝐞.j)/Ũ), j,1..dim), i,1..dim)
 (29)
Type: Expression(Integer)

General Solution

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)::ℒ Symbol);
Type: Matrix(Expression(Integer))
fricas
nrows(J),ncols(J)
 (30)
Type: Tuple(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:

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Ñ:=nullSpace(J)
 (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) )
 (32)
Type: List(Equation(Expression(Integer)))

This defines a family of pre-Frobenius algebras:

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zero? eval(ω,ℰ)
 (33)
Type: Boolean
fricas
Ų:𝐋 := eval(U,ℰ)
 (34)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

Frobenius Form (co-unit)

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d:=ε1*𝐝.1+εi*𝐝.2+εj*𝐝.3+εk*𝐝.4
 (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)))
 (36)
Type: List(Equation(Expression(Integer)))

Express scalar product in terms of Frobenius form

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𝔓:=solve(𝔇,Ξ(sb('p,[i]), i,1..#Ñ)).1
 (37)
Type: List(Equation(Expression(Integer)))
fricas
Ų:=eval(Ų,𝔓)
 (38)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
Y     /
d     =  Ų
 (39)
Type: Boolean

In general the pairing is not symmetric!

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u1:=matrix Ξ(Ξ(retract((𝐞.i 𝐞.j)/Ų), i,1..dim), j,1..dim)
 (40)
Type: Matrix(Expression(Integer))

fricas
)set output algebra on

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)set output tex off

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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

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)set output tex on

The scalar product must be non-degenerate:

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Ů:=determinant u1
 (41)
Type: Expression(Integer)
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factor(numer Ů)/factor(denom Ů)
 (42)
Type: Fraction(Factored(SparseMultivariatePolynomial?(Integer,Kernel(Expression(Integer)))))

Frobenius scalar product of "vectors" and

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a:=sb('a,[1])*i+sb('a,[2])*j
 (43)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
b:=sb('b,[1])*i+sb('b,[2])*j
 (44)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
(a,a)/Ų
 (45)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
(a,b)/Ų
 (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)
 (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))
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eq1:=equate(UXΩ=I);
Type: List(Equation(Expression(Integer)))
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eq2:=equate(ΩXU=I);
Type: List(Equation(Expression(Integer)))
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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))
 (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)
 (49)
Type: Matrix(Expression(Integer))

Check "dimension" and the snake relations.

fricas
O:𝐋:=
Ω    /
Ų
 (50)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
(    I ΩX     )  /
(     Ų I     )  =  I
 (51)
Type: Boolean
fricas
test
(     ΩX I    )  /
(    I Ų      )  =  I
 (52)
Type: Boolean

## Definition 4

Co-algebra

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

fricas
W:=
(Y I) /
Ų
 (53)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
λ:=
(  ΩX I ΩX  ) /
(  I  W  I  )
 (54)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))

fricas
test
(    I ΩX     )  /
(     Y I     )  =  λ
 (55)
Type: Boolean
fricas
test
(     ΩX I    )  /
(    I  Y     )  =  λ
 (56)
Type: Boolean

Co-associativity

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test(
(  λ  ) / _
( I λ ) = _
(  λ  ) / _
( λ I ) )
 (57)
Type: Boolean

fricas
test
e     /
λ     =    ΩX
 (58)
Type: Boolean

Frobenius Condition (fork)

fricas
H :=
Y    /
λ
 (59)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
(   λ I   )  /
(  I Y    )  =  H
 (60)
Type: Boolean
fricas
test
(   I λ   )  /
(    Y I  )  =  H
 (61)
Type: Boolean

Handle

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Φ :=
λ     /
Y
 (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 )
 (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  )
 (64)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test((e,e)/ΦΦ=φφ)
 (65)
Type: Boolean

Bi-algebra conditions

fricas
ΦΦ:=          _
(  λ λ  ) / _
( I X I ) / _
(  Y Y  )
 (66)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test((e,e)/ΦΦ=φφ)
 (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
 (68)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
RS:=
(  Λ Y )/ _
( I Y  )/ _
V
 (69)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LS=RS)
 (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
 (71)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
LD:=
(  I Λ I  ) / _
(   X Y   ) / _
(  I Y    ) / _
V
 (72)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LD=RD)
 (73)
Type: Boolean
fricas
test(RD=RS)
 (74)
Type: Boolean
fricas
test(RD=LS)
 (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
 (76)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
LT:=
( I I Λ  ) / _
(  I Y I ) / _
(   Y I  ) / _
V
 (77)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test(LT=RT)
 (78)
Type: Boolean

The turles are symmetric

fricas
test(RT = X/RT)
 (79)
Type: Boolean
fricas
test(LT = X/LT)
 (80)
Type: Boolean

Five of the six forms are independent.

fricas
test(RT=RS)
 (81)
Type: Boolean
fricas
test(RT=LS)
 (82)
Type: Boolean
fricas
test(RT=RD)
 (83)
Type: Boolean
fricas
test(LT=RS)
 (84)
Type: Boolean
fricas
test(LT=LS)
 (85)
Type: Boolean
fricas
test(LT=RD)
 (86)
Type: Boolean

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