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
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(1) -> <spad>
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)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>
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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: 0.03 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 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 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 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 SEC.
compiling exported hyper : List % -> %
Time: 0 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: (Hashable)
****** 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: 95989)
;;; *** |CaleyDickson| REDEFINED
;;; *** |CaleyDickson| REDEFINED
Time: 0.12 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: 0.18 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
; compiling file "/var/aw/var/LatexWiki/CALEY.NRLIB/CALEY.lsp" (written 02 DEC 2024 03:01:39 AM):
; wrote /var/aw/var/LatexWiki/CALEY.NRLIB/CALEY.fasl
; compilation finished in 0:00:00.068
------------------------------------------------------------------------
CaleyDickson is now explicitly exposed in frame initial
CaleyDickson will be automatically loaded when needed from
/var/aw/var/LatexWiki/CALEY.NRLIB/CALEY
Test
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R := FRAC POLY Integer
Type: Type
Complex Numbers
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C := CaleyDickson(R,'i,1)
Type: Type
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rank()$C
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Ce:ILIST(C,0) := construct entries basis()$C
Type: IndexedList
?(
CaleyDickson(Fraction(Polynomial(Integer)),
i,
1),
0)
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matrix [[Ce.i * Ce.j for j in 0..#Ce-1] for i in 0..#Ce-1]
Type: Matrix(
CaleyDickson(Fraction(Polynomial(Integer)),
i,
1))
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--
-- compare
--
Cg:ILIST(Complex R,0) := construct map(x+-> complex(x.1,x.2),
1$SquareMatrix(2,FRAC INT)::List List FRAC INT)
Type: IndexedList
?(Complex(Fraction(Polynomial(Integer))),
0)
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matrix [[Cg.i * Cg.j for j in 0..#Cg-1] for i in 0..#Cg-1]
Type: Matrix(Complex(Fraction(Polynomial(Integer))))
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-- normed?
c1:C := hyper [subscript('c1,[i]) for i in 1..2]
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c2:C := hyper [subscript('c2,[i]) for i in 1..2]
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c3:C := hyper [subscript('c3,[i]) for i in 1..8]
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-- Normed?
test(norm(c1*c2)=norm(c1)*norm(c2))
Type: Boolean
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-- Commutative?
test( c1 * c2 = c2 * c1 )
Type: Boolean
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-- Associative?
test((c1 * c2) * c3 = c1 * (c2 * c3))
Type: Boolean
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-- inverse
c1inv := inv c1
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test(c1 * c1inv = 1)
Type: Boolean
Quaternions
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Q := CaleyDickson(C,'j,1)
Type: Type
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rank()$Q
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Qe:ILIST(Q,0) := construct entries basis()$Q
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matrix [[Qe.i * Qe.j for j in 0..#Qe-1] for i in 0..#Qe-1]
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--
-- compare
--
Qg:ILIST(Quaternion R,0) := construct map(x+-> quatern(x.1,x.2,x.3,x.4),
1$SquareMatrix(4,FRAC INT)::List List FRAC INT)
Type: IndexedList
?(Quaternion(Fraction(Polynomial(Integer))),
0)
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matrix [[Qg.i * Qg.j for j in 0..#Qg-1] for i in 0..#Qg-1]
Type: Matrix(Quaternion(Fraction(Polynomial(Integer))))
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q1:Q := hyper [subscript('q1,[i]) for i in 1..4]
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q2:Q := hyper [subscript('q2,[i]) for i in 1..4]
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q3:Q := hyper [subscript('q3,[i]) for i in 1..8]
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-- Normed?
test( norm(q1*q2) = norm(q1)*norm(q2))
Type: Boolean
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-- Commutative?
test( q1 * q2 = q2 * q1 )
Type: Boolean
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-- Associative?
test((q1 * q2) * q3 = q1 * (q2 * q3))
Type: Boolean
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-- inverse
q1inv := inv q1
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test(q1 * q1inv = 1)
Type: Boolean
Octonions
Ref: http://en.wikipedia.org/wiki/Octonion
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O:=CaleyDickson(Q,'k,1)
Type: Type
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rank()$O
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Oe:ILIST(O,0) := construct entries basis()$O
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matrix [[Oe.i * Oe.j for j in 0..#Oe-1] for i in 0..#Oe-1]
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--
-- compare
--
Og:ILIST(Octonion R,0):=map(x+-> octon(x.1,x.2,x.3,x.4,x.5,x.6,x.7,x.8),
1$SquareMatrix(8,FRAC INT)::List List FRAC INT)
Type: IndexedList
?(Octonion(Fraction(Polynomial(Integer))),
0)
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matrix [[Og.i * Og.j for j in 0..#Og-1] for i in 0..#Og-1]
Type: Matrix(Octonion(Fraction(Polynomial(Integer))))
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o1:O := hyper [subscript('o1,[i]) for i in 1..8]
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o2:O := hyper [subscript('o2,[i]) for i in 1..8]
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o3:O := hyper [subscript('o3,[i]) for i in 1..8]
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-- normed?
test(norm(o1*o2)=norm(o1)*norm(o2))
Type: Boolean
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-- Commutative?
test( o1 * o2 = o2 * o1 )
Type: Boolean
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-- Associative?
test((o1 * o2) * o3 = o1 * (o2 * o3))
Type: Boolean
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-- Alternative?
test((o1 * o2) * o1 = o1 * (o2 * o1))
Type: Boolean
Split-Octonions
Ref: http://en.wikipedia.org/wiki/Split-octonion
Note: Our table below is not identical the one shown in the reference where a different convention is used to define multiplication.
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sO:=CaleyDickson(Q,'ℓ,-1)
Type: Type
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rank()$sO
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sOe:ILIST(sO,0) := construct entries basis()$sO
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matrix [[sOe.i * sOe.j for j in 0..#sOe-1] for i in 0..#sOe-1]
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so1:sO := hyper [subscript('so1,[i]) for i in 1..8]
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so2:sO := hyper [subscript('so2,[i]) for i in 1..8]
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so3:sO := hyper [subscript('so3,[i]) for i in 1..8]
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-- Normed?
test(norm(so1*so2)=norm(so1)*norm(so2))
Type: Boolean
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-- Commutative?
test( so1 * so2 = so2 * so1 )
Type: Boolean
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-- Associative?
test((so1 * so2) * so3 = so1 * (so2 * so3))
Type: Boolean
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-- Alternative?
test((so1 * so2) * so1 = so1 * (so2 * so1))
Type: Boolean
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-- inverse
so1inv := inv so1;
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test(so1 * so1inv = 1)
Type: Boolean
Sedenions
Ref: http://en.wikipedia.org/wiki/Sedenion
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S:=CaleyDickson(O,'l,1)
Type: Type
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rank()$S
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Se:ILIST(S,0) := construct entries basis()$S
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matrix [[Se.i * Se.j for j in 0..#Se-1] for i in 0..#Se-1]
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s1:S := hyper [subscript('s1,[i]) for i in 1..16]
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s2:S := hyper [subscript('s2,[i]) for i in 1..16]
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s3:S := hyper [subscript('s3,[i]) for i in 1..16]
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-- normed?
test(norm(s1*s2)=norm(s1)*norm(s2))
Type: Boolean
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-- Commutative
test( s1 * s2 = s2 * s1 )
Type: Boolean
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-- Associative?
test( (s1 * s2) * s3 = s1 * (s2 * s3) )
Type: Boolean
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-- Alternative?
test((s1 * s2) * s1 = - s1 * (s2 * s1))
Type: Boolean
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-- zero divisor
(Se.3 + Se.10) * (Se.6 - Se.15)
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-- inverse
s1inv := inv s1;
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test(s1 * s1inv = 1)
Type: Boolean
Power Associative?
Algebra with associative powers
Ref: http://eom.springer.de/a/a011410.htm
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test( s1^2 * s1 = s1 * s1^2 )
Type: Boolean
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test( (s1^2 * s1) * s1 = s1^2 * s1^2 )
Type: Boolean
Conversions
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test(s1=hyper scalars s1)
Type: Boolean
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scalars(s1)::List Symbol
Type: List(Symbol)