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Quaternion Algebra Is Frobenius In Many Ways

Linear operators over a 4-dimensional vector space representing quaternion algebra

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
-- 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 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:List ℒ      := basisOut()
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Now generate structure constants for Quaternion Algebra

The basis consists of the real and imaginary units. We use quaternion multiplication to form the "multiplication table" as a matrix. Then the structure constants can be obtained by dividing each matrix entry by the list of basis vectors.

Split-complex and co-quaternions can be specified by Caley-Dickson parameters (i2, j2)

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

\label{eq3}i^{2}(3)
Type: Symbol
fricas
--i2:= -1  -- complex
j2:=sp('j,[2])

\label{eq4}j^{2}(4)
Type: Symbol
fricas
--j2:= -1  -- quaternion
k2:=-i2*j2;
Type: Polynomial(Integer)
fricas
QQ := ℂ(ℂ(ℚ,'i,-i2),'j,-j2);
Type: Type

Basis: Each B.i is a quaternion number

fricas
B:List QQ := map(x +-> hyper x,1$SQMATRIX(dim,ℚ)::List List ℚ)

\label{eq5}\left[ 1, \: i , \: j , \:{ij}\right](5)
Type: List(CaleyDickson(CaleyDickson(Expression(Integer),i,-(i[;2])),j,-(j[;2])))
fricas
-- Multiplication table:
M:Matrix QQ := matrix [[B.i*B.j for i in 1..dim] for j in 1..dim]

\label{eq6}\left[ 
\begin{array}{cccc}
1 & i & j &{ij}
\
i &{i^{2}}& -{ij}&{-{i^{2}}j}
\
j &{ij}&{j^{2}}&{{j^{2}}i}
\
{ij}&{{i^{2}}j}&{-{j^{2}}i}& -{{i^{2}}\ {j^{2}}}
(6)
Type: Matrix(CaleyDickson(CaleyDickson(Expression(Integer),i,-(i[;2])),j,-(j[;2])))
fricas
-- Function to divide the matrix entries by a basis element
S(y) == map(x +-> real real(x/y),M)
Type: Void
fricas
-- The result is a nested list
ѕ :=map(S,B)::List List List ℚ
fricas
Compiling function S with type CaleyDickson(CaleyDickson(Expression(
      Integer),i,-(i[;2])),j,-(j[;2])) -> Matrix(Expression(Integer))

\label{eq7}\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] (7)
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)
There are no library operations named ⅆ Use HyperDoc Browse or issue )what op ⅆ to learn if there is any operation containing " ⅆ " in its name. Cannot find a definition or applicable library operation named ⅆ with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Units

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

Multiplication of arbitrary quaternions a and b

fricas
a:=Σ(sb('a,[i])*ⅇ.i, i,1..dim)
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Multiplication is Associative

fricas
test(
  ( I Y ) / _
  (  Y  ) = _
  ( Y I ) / _
  (  Y  ) )
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) Variable(Y)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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)
There are no library operations named ⅆ Use HyperDoc Browse or issue )what op ⅆ to learn if there is any operation containing " ⅆ " in its name. Cannot find a definition or applicable library operation named ⅆ with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need. FriCAS will attempt to step through and interpret the code. There are no library operations named ⅆ Use HyperDoc Browse or issue )what op ⅆ to learn if there is any operation containing " ⅆ " in its name.
Cannot find a definition or applicable library operation named ⅆ with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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{eq8}
  \Phi = \{ \phi^{ijk} = {y^e}_{ij} u_{ek} - u_{ie} {y_e}^{jk} \}
  (8)
(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
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(Λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

forms a non-degenerate associative scalar product for Y

fricas
Ũ := Ù

\label{eq9}��(9)
Type: Variable(Ù)
fricas
test
     (    Y I    )  /
           Ũ        =
     (    I Y    )  /
           Ũ
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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)::List Symbol);
There are 2 exposed and 0 unexposed library operations named ravel having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ravel to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel with argument type(s) Variable(U)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

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)
There are 3 exposed and 3 unexposed library operations named nullSpace having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op nullSpace to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named nullSpace with argument type(s) Variable(J)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

This defines a family of pre-Frobenius algebras:

fricas
zero? eval(ω,ℰ)
There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)) Variable(ℰ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Frobenius Form (co-unit)

fricas
d:=ε1*ⅆ.1+εi*ⅆ.2+εj*ⅆ.3+εk*ⅆ.4
There are no library operations named ⅆ Use HyperDoc Browse or issue )what op ⅆ to learn if there is any operation containing " ⅆ " in its name.
Cannot find a definition or applicable library operation named ⅆ with argument type(s) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Express scalar product in terms of Frobenius form

fricas
Ξ:=solve(%,[sb('p,[i]) for i in 1..#Ñ]).1
There are 2 exposed and 1 unexposed library operations named # having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op # to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named # with argument type(s) Variable(Ñ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

In general the pairing is not symmetric!

fricas
u1:=matrix [[retract((ⅇ.i ⅇ.j)/Ų) for i in 1..dim] for j in 1..dim]
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

The scalar product must be non-degenerate:

fricas
Ů:=determinant u1
There are 3 exposed and 1 unexposed library operations named determinant having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op determinant to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named determinant with argument type(s) Variable(u1)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing is a special case

fricas
ck:=solve(equate(eval(Ũ,Ξ)=Ų),[ε1,εi,εj,εk]).1
There are no library operations named equate Use HyperDoc Browse or issue )what op equate to learn if there is any operation containing " equate " in its name.
Cannot find a definition or applicable library operation named equate with argument type(s) Equation(Expression(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Frobenius scalar product of "vector" quaternions a and b

fricas
a:=sb('a,[1])*i+sb('a,[2])*j
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(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)
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

The common demoninator is 1/\sqrt{\mathring{U}}

fricas
squareFreePart factor denom Ů / squareFreePart factor numer Ů
Function: squareFree : % -> Factored(%) is missing from domain: Factored(SparseMultivariatePolynomial(Integer,Kernel(Expression(Integer)))) Internal Error The function squareFree with signature (Factored $)$ is missing from domain Factored (SparseMultivariatePolynomial (Integer) (Kernel (Expression (Integer))))

Check "dimension" and the snake relations.

fricas
O:ℒ:=
       Ω    /
       Ų

\label{eq10}�� \  ��(10)
Type: ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer))
fricas
test
    (    I ΩX     )  /
    (     Ų I     )  =  I
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) Variable(ΩX)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing co-scalar

fricas
eval(Ω,ck)
There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Definition 4

Co-algebra

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

fricas
W:=
  (Y I) /
    Ų
There are no exposed library operations named Y but there are 2 unexposed operations with that name. Use HyperDoc Browse or issue )display op Y to learn more about the available operations.
Cannot find a definition or applicable library operation named Y with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Cartan-Killing co-multiplication

fricas
eval(λ,ck)

\label{eq11}��(11)
Type: Expression(Integer)

fricas
test
     (    I ΩX     )  /
     (     Y I     )  =  λ
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) Variable(ΩX)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Co-associativity

fricas
test(
  (  λ  ) / _
  ( I λ ) = _
  (  λ  ) / _
  ( λ I ) )
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) Variable(λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

fricas
test
         q     /
         λ     =    ΩX
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

Frobenius Condition (fork)

fricas
H :=
         Y    /
         λ

\label{eq12}Y \over ��(12)
Type: Fraction(Polynomial(Integer))
fricas
test
     (   λ I   )  /
     (  I Y    )  =  H
There are no library operations named λ Use HyperDoc Browse or issue )what op λ to learn if there is any operation containing " λ " in its name.
Cannot find a definition or applicable library operation named λ with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

The Cartan-Killing form makes H of the Frobenius condition idempotent

fricas
test( eval(H,ck)=eval(H/H,ck) )

\label{eq13} \mbox{\rm false} (13)
Type: Boolean

But it is not unique. E.g. other idempots

fricas
h1:=map(numer,ravel(H-H/H)::List FRAC POLY INT);
There are 2 exposed and 0 unexposed library operations named ravel having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ravel to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel with argument type(s) Fraction(Polynomial(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Handle and handle element

fricas
Φ :=
         λ     /
         Y

\label{eq14}�� \over Y(14)
Type: Fraction(Polynomial(Integer))
fricas
Φ1 :=    q     /
         Φ
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)

The Cartan-Killing form makes Φ into the identity

fricas
test( eval(Φ,ck)=I )

\label{eq15} \mbox{\rm false} (15)
Type: Boolean
fricas
eval(Φ1,ck)

\label{eq16}�� 1(16)
Type: Expression(Integer)

but it can be the identity in many ways. For example,

fricas
solve(equate(eval(Φ,[ε1=1,εi=1,εj=1,j2=-1])=I),[εk])
There are no library operations named equate Use HyperDoc Browse or issue )what op equate to learn if there is any operation containing " equate " in its name.
Cannot find a definition or applicable library operation named equate with argument type(s) Equation(Fraction(Polynomial(Integer)))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

If handle is identity then fork is idempotent but the converse is not true

fricas
Φ1:=map(numer,ravel(Φ-I)::List FRAC POLY INT);
There are 2 exposed and 0 unexposed library operations named ravel having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op ravel to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named ravel with argument type(s) Fraction(Polynomial(Integer))
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Figure 12

fricas
φφ:=          _
  ( Ω  Ω  ) / _
  ( X I I ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y );
There are no exposed library operations named I but there is one unexposed operation with that name. Use HyperDoc Browse or issue )display op I to learn more about the available operation.
Cannot find a definition or applicable library operation named I with argument type(s) Variable(I)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

For Cartan-Killing this is just the co-scalar

fricas
test(eval(φφ,ck)=eval(Ω,ck))
There are 12 exposed and 6 unexposed library operations named eval having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op eval to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation.
Cannot find a definition or applicable library operation named eval with argument type(s) ClosedLinearOperator(OrderedVariableList([1,i,j,k]),Expression(Integer)) Variable(ck)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Bi-algebra conditions

fricas
ΦΦ:=          _
  (  λ λ  ) / _
  ( I I X ) / _
  ( I X I ) / _
  ( I I X ) / _
  (  Y  Y ) ;
There are no library operations named λ Use HyperDoc Browse or issue )what op λ to learn if there is any operation containing " λ " in its name.
Cannot find a definition or applicable library operation named λ with argument type(s) Variable(λ)
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.

Theorem 8.3

fricas
u2:=map(retract,matrix [ _
        [q/d,     i/d,    j/d,     k/d], _
        [i/d,  i2*q/d,    k/d,  i2*j/d], _
        [j/d,    -k/d, j2*q/d, -j2*i/d], _
        [k/d, -i2*j/d, j2*i/d, k2*q/d]])
Function: setelt : (%,Integer,Integer) -> Integer is missing from domain: Vector(Integer) Internal Error The function setelt with signature (Integer)$(Integer)(Integer) is missing from domain Vector(Integer)




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