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	The Algebra of Complex Numbers Is Frobenius In Many Ways
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CaleyDickson

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Editor: Bill Page Time: 2011/04/15 15:59:52 GMT-7
Note: new

changed:
-
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, ...

\begin{spad}
)abbrev domain CALEY CaleyDickson
CaleyDickson(R:CommutativeRing,C:ComplexCategory(R)):Exports == Implementation where
  Exports ==> ComplexCategory(C) with
    if C has Field then Field
  Implementation ==> add
    Rep == DirectProduct(2,C)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    pair(x:C,y:C):Rep == directProduct vector [x,y]

    0:% == per pair(0,0)
    zero?(x):Boolean == rep(x)=0
    1:% == per pair(1,0)
    imaginary():% == per pair(0,1)
    basis():Vector % == vector [1,imaginary()]
    complex(x:C,y:C):% == per pair(x,y)
    real(x:%):C == rep(x).1
    imag(x:%):C == rep(x).2

    (x:% = y:%):Boolean == rep x = rep y
    (x:% * y:%):% == per pair(real x * real y - conjugate imag y * imag x, imag y * real x + imag x * conjugate real y)
    (x:% + y:%):% == per(rep x + rep y)
    (x:% - y:%):% == per(rep x - rep y)
    conjugate(x:%):% == per pair(conjugate(real x), -imag x)
    if C has Field then
      inv(x:%):% == per(inv(real(conjugate x * x))$C * rep conjugate x)
      (x:% / y:%):% == x * inv(y)
    retract(x:%):C ==
       imag x ~=0 => error "not retractable"
       real x
    coerce(x:C):% == per pair(x,0)
    coerce(x:Integer):% == per pair(x::R::C,0)
    coerce(x:%):OutputForm == real(x)::OutputForm + message("%I")*imag(x)::OutputForm
\end{spad}

\begin{axiom}
Q:=CaleyDickson(FRAC INT,Complex FRAC INT)
q:Q:=complex(complex(1,1),complex(1,1))
\end{axiom}

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, ...

spad

)abbrev domain CALEY CaleyDickson
CaleyDickson(R:CommutativeRing,C:ComplexCategory(R)):Exports == Implementation where
  Exports ==> ComplexCategory(C) with
    if C has Field then Field
  Implementation ==> add
    Rep == DirectProduct(2,C)
    rep(x:%):Rep == x pretend Rep
    per(x:Rep):% == x pretend %

    pair(x:C,y:C):Rep == directProduct vector [x,y]

    0:% == per pair(0,0)
    zero?(x):Boolean == rep(x)=0
    1:% == per pair(1,0)
    imaginary():% == per pair(0,1)
    basis():Vector % == vector [1,imaginary()]
    complex(x:C,y:C):% == per pair(x,y)
    real(x:%):C == rep(x).1
    imag(x:%):C == rep(x).2

    (x:% = y:%):Boolean == rep x = rep y
    (x:% * y:%):% == per pair(real x * real y - conjugate imag y * imag x, imag y * real x + imag x * conjugate real y)
    (x:% + y:%):% == per(rep x + rep y)
    (x:% - y:%):% == per(rep x - rep y)
    conjugate(x:%):% == per pair(conjugate(real x), -imag x)
    if C has Field then
      inv(x:%):% == per(inv(real(conjugate x * x))$C * rep conjugate x)
      (x:% / y:%):% == x * inv(y)
    retract(x:%):C ==
       imag x ~=0 => error "not retractable"
       real x
    coerce(x:C):% == per pair(x,0)
    coerce(x:Integer):% == per pair(x::R::C,0)
    coerce(x:%):OutputForm == real(x)::OutputForm + message("%I")*imag(x)::OutputForm

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/6639839417993251745-25px001.spad using 
      old system compiler.
   CALEY abbreviates domain CaleyDickson 
------------------------------------------------------------------------
   initializing NRLIB CALEY for CaleyDickson 
   compiling into NRLIB CALEY 
   compiling local rep : $ -> DirectProduct(2,C)
      CALEY;rep is replaced by x 
Time: 0.41 SEC.

   compiling local per : DirectProduct(2,C) -> $
      CALEY;per is replaced by x 
Time: 0.01 SEC.

   compiling local pair : (C,C) -> DirectProduct(2,C)
Time: 0.02 SEC.

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

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

   compiling exported One : () -> $
Time: 0.01 SEC.

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

   compiling exported basis : () -> Vector $
Time: 0.09 SEC.

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

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

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

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

   compiling exported * : ($,$) -> $
Time: 0.01 SEC.

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

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

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

****** Domain: C already in scope
augmenting C: (Field)
   compiling exported inv : $ -> $
Time: 0.08 SEC.

   compiling exported / : ($,$) -> $
Time: 0.01 SEC.

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

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

   compiling exported coerce : Integer -> $
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

****** Domain: C already in scope
augmenting C: (EuclideanDomain)
****** Domain: C already in scope
augmenting C: (PolynomialFactorizationExplicit)
****** 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: (EuclideanDomain)
****** 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: (LinearlyExplicitRingOver (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:  3040)

;;;     ***       |CaleyDickson| REDEFINED

;;;     ***       |CaleyDickson| REDEFINED
Time: 3.82 SEC.

   Cumulative Statistics for Constructor CaleyDickson
      Time: 4.47 seconds

   finalizing NRLIB CALEY 
   Processing CaleyDickson for Browser database:
--->-->CaleyDickson(): Missing Description
; compiling file "/var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY.lsp" (written 15 APR 2011 03:59:58 PM):

; /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY.fasl written
; compilation finished in 0:00:00.522
------------------------------------------------------------------------
   CaleyDickson is now explicitly exposed in frame initial 
   CaleyDickson will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CALEY.NRLIB/CALEY

   >> System error:
   The bounding indices 163 and 162 are bad for a sequence of length 162.
See also:
  The ANSI Standard, Glossary entry for "bounding index designator"
  The ANSI Standard, writeup for Issue SUBSEQ-OUT-OF-BOUNDS:IS-AN-ERROR

axiom

Q:=CaleyDickson(FRAC INT,Complex FRAC INT)

$\label{eq1}\hbox{\axiomType{CaleyDickson}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ }) , \hbox{\axiomType{Complex}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })))$

(1)

Type: Type

axiom

q:Q:=complex(complex(1,1),complex(1,1))

$\label{eq2}1 + i +{\%I \ {\left(1 + i \right)}}$

(2)

Type: CaleyDickson?(Fraction(Integer),Complex(Fraction(Integer)))