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	The Algebra of Complex Numbers Is Frobenius In Many Ways
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CaleyDickson

Edit detail for CaleyDickson revision 16 of 18

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Editor: Bill Page Time: 2013/04/01 21:18:41 GMT+0
Note: Rep == vs :=

Ref:

http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction

"The Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by this process are known as Cayley–Dickson algebras; since they extend the complex numbers, ... "

complex numbers, quaternions, octonions, sedenions, ...

http://en.wikipedia.org/wiki/Hypercomplex_number

fricas

(1) -> <spad>

fricas

)abbrev domain CALEY CaleyDickson
CaleyDickson(C:CommutativeRing,gen:Symbol,gamma:C):ComplexCategory(C) with
    hyper:List % -> %
      ++ convert a list of scalars to a hyper-comnplex number
    scalars: % -> List %
      ++ convert a hyper-complex number to a list of scalars
  == add
    Rep := Record(re:C, im:C)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    complex(x:C,y:C):% == per [x,y]
    real(x:%):C == rep(x).re
    imag(x:%):C == rep(x).im
    --mul(x:C,y:%):% == complex(x * real y, x * imag y)
    mul(x:C,y:%):% == per [x * rep(y).re, x * rep(y).im]

    -- Many funtctions are inherited from ComplexCategory
    --
    -- To Do:
    -- 1) Check which functions are still correct for higher-order algebras!

    0:% == complex(0,0)
    --zero?(x:%):Boolean == zero? real x and zero? imag x
    zero?(x:%):Boolean == x = 0
    1:% == complex(1,0)
    --one?(x:%):Boolean  == one? real x and zero? imag x
    one?(x:%):Boolean == x = 1

    if C has conjugate:C->C then
      -- In general we need conjugate
      (x:% * y:%):% ==
        complex(real x * real y - gamma*conjugate imag y * imag x,
                imag y * real x + imag x * conjugate real y)
      conjugate(x:%):% == complex(conjugate(real x), -imag x)
    else
      -- If not complex then conjugate is identity
      (x:% * y:%):% ==
        complex(real x * real y - gamma*imag y * imag x,
                imag y * real x + imag x * real y)
      conjugate(x:%):% == complex(real x, -imag x)

    -- correct order
    norm(x:%):C == retract(conjugate(x)*x)

    if C has Field then
      inv(x:%):% == mul(inv norm x, conjugate x)
      (x:% / y:%):% == x * inv(y)

    if C has rank:()->PositiveInteger then
      rank():PositiveInteger == 2*rank()$C
    else
      rank():PositiveInteger == 2

    if C has basis:()->Vector C then
      basis():Vector % ==
        concat([complex(i,0) for i in entries basis()$C],
               [complex(0,i) for i in entries basis()$C])
    else
      basis():Vector % == [1,imaginary()]

    if C has scalars: C -> List C then
      scalars(x:%):List % == map(coerce,concat(scalars real x, scalars imag x))$ListFunctions2(C,%)
    else
      scalars(x:%):List % == [ coerce real x, coerce imag x ]

    if C has hyper:List C -> C then
      hyper(x:List %):% ==
        h:Integer := divide(#x,2).quotient
        complex(hyper([retract(x.i)@C for i in 1..h]),hyper([retract(x.i)@C for i in h+1..#x]))
    else
      hyper(x:List %):% == complex(retract x.1,retract x.2)

    coerce(x:%):OutputForm ==
      outr:=real(x)::OutputForm
      imag x = 0 => return outr
      outi := hconcat(imag(x)::OutputForm, gen::OutputForm)
      if imag x = 1 then
        outi := gen::OutputForm
      if imag x = -1 then
        outi := -(gen::OutputForm)
      if C has imaginary:()->C then
        if imag x = -imaginary()$C then
          outi := -hconcat(imaginary()$C::OutputForm,gen::OutputForm)
      real x = 0 => return outi
      return outr + outi</spad>

fricas

Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/381330559998884003-25px001.spad
      using old system compiler.
   CALEY abbreviates domain CaleyDickson 
------------------------------------------------------------------------
   initializing NRLIB CALEY for CaleyDickson 
   compiling into NRLIB CALEY 
****** Domain: C already in scope
   compiling local rep : $ -> Rep
      CALEY;rep is replaced by x 
Time: 0.07 SEC.

   compiling local per : Rep -> $
      CALEY;per is replaced by x 
Time: 0 SEC.

   compiling exported complex : (C,C) -> $
Time: 0 SEC.

   compiling exported real : $ -> C
Time: 0 SEC.

   compiling exported imag : $ -> C
Time: 0 SEC.

   compiling local mul : (C,$) -> $
Time: 0 SEC.

   compiling exported Zero : () -> $
Time: 0 SEC.

   compiling exported zero? : $ -> Boolean
Time: 0 SEC.

   compiling exported One : () -> $
Time: 0 SEC.

   compiling exported one? : $ -> Boolean
Time: 0 SEC.

augmenting C: (SIGNATURE C conjugate (C C))
   compiling exported * : ($,$) -> $
Time: 0.01 SEC.

   compiling exported conjugate : $ -> $
Time: 0 SEC.

   compiling exported * : ($,$) -> $
Time: 0 SEC.

   compiling exported conjugate : $ -> $
Time: 0 SEC.

   compiling exported norm : $ -> C
Time: 0 SEC.

****** Domain: C already in scope
augmenting C: (Field)
   compiling exported inv : $ -> $
Time: 0 SEC.

   compiling exported / : ($,$) -> $
Time: 0 SEC.

augmenting C: (SIGNATURE C rank ((PositiveInteger)))
   compiling exported rank : () -> PositiveInteger
Time: 0.01 SEC.

   compiling exported rank : () -> PositiveInteger
      CALEY;rank;Pi;19 is replaced by 2 
Time: 0 SEC.

augmenting C: (SIGNATURE C basis ((Vector C)))
   compiling exported basis : () -> Vector $
Time: 0.01 SEC.

   compiling exported basis : () -> Vector $
Time: 0 SEC.

augmenting C: (SIGNATURE C scalars ((List C) C))
   compiling exported scalars : $ -> List $
Time: 0 SEC.

   compiling exported scalars : $ -> List $
Time: 0 SEC.

augmenting C: (SIGNATURE C hyper (C (List C)))
   compiling exported hyper : List $ -> $
Time: 0.01 SEC.

   compiling exported hyper : List $ -> $
Time: 0.01 SEC.

   compiling exported coerce : $ -> OutputForm
augmenting C: (SIGNATURE C imaginary (C))
Time: 0 SEC.

****** Domain: C already in scope
augmenting C: (RadicalCategory)
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
****** Domain: C already in scope
augmenting C: (RealNumberSystem)
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
****** Domain: C already in scope
augmenting C: (Comparable)
****** Domain: C already in scope
augmenting C: (ConvertibleTo (InputForm))
****** Domain: C already in scope
augmenting C: (ConvertibleTo (Pattern (Float)))
****** Domain: C already in scope
augmenting C: (ConvertibleTo (Pattern (Integer)))
****** Domain: C already in scope
augmenting C: (DifferentialRing)
****** Domain: C already in scope
augmenting C: (Eltable C C)
****** Domain: C already in scope
augmenting C: (Evalable C)
****** Domain: C already in scope
augmenting C: (Field)
****** Domain: C already in scope
augmenting C: (Finite)
****** Domain: C already in scope
augmenting C: (FiniteFieldCategory)
****** Domain: C already in scope
augmenting C: (InnerEvalable (Symbol) C)
****** Domain: C already in scope
augmenting C: (IntegerNumberSystem)
****** Domain: C already in scope
augmenting C: (IntegralDomain)
****** Domain: C already in scope
augmenting C: (LinearlyExplicitOver (Integer))
****** Domain: C already in scope
augmenting C: (PartialDifferentialRing (Symbol))
****** Domain: C already in scope
augmenting C: (PatternMatchable (Float))
****** Domain: C already in scope
augmenting C: (PatternMatchable (Integer))
****** Domain: C already in scope
augmenting C: (RealConstant)
****** Domain: C already in scope
augmenting C: (RealNumberSystem)
****** Domain: C already in scope
augmenting C: (RetractableTo (Fraction (Integer)))
****** Domain: C already in scope
augmenting C: (RetractableTo (Integer))
****** Domain: C already in scope
augmenting C: (TranscendentalFunctionCategory)
(time taken in buildFunctor:  270)

;;;     ***       |CaleyDickson| REDEFINED

;;;     ***       |CaleyDickson| REDEFINED
Time: 0.33 SEC.

   Warnings: 
      [1] rep: pretendRep -- should replace by @
      [2] per: pretend$ -- should replace by @
      [3] real:  re has no value
      [4] imag:  im has no value
      [5] mul:  re has no value
      [6] mul:  im has no value
      [7] scalars:  $$ has no value

   Cumulative Statistics for Constructor CaleyDickson
      Time: 0.45 seconds

   finalizing NRLIB CALEY 
   Processing CaleyDickson for Browser database:
--->-->CaleyDickson(constructor): Not documented!!!!
--------(hyper (% (List %)))---------
--->-->CaleyDickson((hyper (% (List %)))): Improper first word in comments: convert
"convert a list of scalars to a hyper-comnplex number"
--------(scalars ((List %) %))---------
--->-->CaleyDickson((scalars ((List %) %))): Improper first word in comments: convert
"convert a hyper-complex number to a list of scalars"
--->-->CaleyDickson(): Missing Description

   >> System error:
   failed to find the TRUENAME of /var/aw/var/LatexWiki/CALEY.erlib:
  No such file or directory

Test

fricas

R := FRAC POLY Integer

$\label{eq1}\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ }))$

(1)

Type: Type

Complex Numbers

fricas

C := CaleyDickson(R,'i,1)

   CaleyDickson is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Quaternions

fricas

Q := CaleyDickson(C,'j,1)

   CaleyDickson is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Octonions

Ref: http://en.wikipedia.org/wiki/Octonion

fricas

O:=CaleyDickson(Q,'k,1)

   CaleyDickson is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Split-Octonions

Ref: http://en.wikipedia.org/wiki/Split-octonion

Note: Our table below is not identical the one shown in the reference where a different convention is used to define multiplication.

fricas

sO:=CaleyDickson(Q,'ℓ,-1)

   CaleyDickson is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Sedenions

Ref: http://en.wikipedia.org/wiki/Sedenion

fricas

S:=CaleyDickson(O,'l,1)

   CaleyDickson is an unknown constructor and so is unavailable. Did 
      you mean to use -> but type something different instead?

Power Associative?

Algebra with associative powers

Ref: http://eom.springer.de/a/a011410.htm

fricas

test( s1^2 * s1 = s1 * s1^2 )

$\label{eq2} \mbox{\rm true}$

(2)

Type: Boolean

fricas

test( (s1^2 * s1) * s1 = s1^2 * s1^2 )

$\label{eq3} \mbox{\rm true}$

(3)

Type: Boolean

Conversions

fricas

test(s1=hyper scalars s1)

   There are no library operations named scalars 
      Use HyperDoc Browse or issue
                              )what op scalars
      to learn if there is any operation containing " scalars " in its 
      name.

   Cannot find a definition or applicable library operation named 
      scalars with argument type(s) 
                                Variable(s1)

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.