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Submitted by : (unknown) at: 2007-11-17T21:53:49-08:00 (9 years ago)
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originally posted by Anonymous

I've been checking quaternion support in Axiom. And I've found out that it is a bit too restricted. More exactly it seems to be aimed on Hamiltonian quaternions only:

fricas
q := quatern(0,1,0,0)

\label{eq1}i(1)
Type: Quaternion(Integer)
fricas
q^2

\label{eq2}- 1(2)
Type: Quaternion(Integer)

This is fine over the reals. But only there. On any field with bigger group of square classes (e.g. rationals, algebraic rationals,...) we have far more quaternion algebras. Namely for any two square classes a, b there exist a quaternion algebra with i^2 = a and j^2 = b.

Is there a way to declare such a quaternions in Axiom?

Yes, use GeneralQuaternion?:

fricas
qD := GeneralQuaternion(Fraction(Polynomial(Integer)), a, b)

\label{eq3}\hbox{\axiomType{GeneralQuaternion}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })) , a , b)(3)
Type: Type
fricas
quatern(0, 1, 0, 0)$qD^2

\label{eq4}a(4)
Type: GeneralQuaternion?(Fraction(Polynomial(Integer)),a,b)
fricas
quatern(0, 0, 1, 0)$qD^2

\label{eq5}b(5)
Type: GeneralQuaternion?(Fraction(Polynomial(Integer)),a,b)
fricas
quatern(0, 0, 0, 1)$qD^2

\label{eq6}-{a \  b}(6)
Type: GeneralQuaternion?(Fraction(Polynomial(Integer)),a,b)

From the description of quatern on in section
9.64 Quaternion, page 734 of the Axiom Book I think that what you want is not possible with this domain. But since the quaternions can be thought of as a Clifford algebra, please refer to section 9.10.2 The Quaternion Numbers as a Clifford Algebra, page 483 of the Book For example
fricas
K := Fraction Polynomial Integer

\label{eq7}\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ }))(7)
Type: Type
fricas
m := matrix [ [a,0],[0,b] ]

\label{eq8}\left[ 
\begin{array}{cc}
a & 0 
\
0 & b 
(8)
Type: Matrix(Polynomial(Integer))
fricas
H := CliffordAlgebra(2, K, m)

\label{eq9}\hbox{\axiomType{CliffordAlgebra}\ } (2, \hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Integer}\ })) , [ [ a , 0 ] , [ 0, b ] ])(9)
Type: Type
fricas
i: H := e(1)

\label{eq10}e_{1}(10)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])
fricas
j: H := e(2)

\label{eq11}e_{2}(11)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])
fricas
k: H := i * j

\label{eq12}{e_{1}}\ {e_{2}}(12)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])
fricas
i^2

\label{eq13}a(13)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])
fricas
j^2

\label{eq14}b(14)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])
fricas
k^2

\label{eq15}-{a \  b}(15)
Type: CliffordAlgebra(2,Fraction(Polynomial(Integer)),[[a,0],[0,b]])

property change
Tue, 08 Mar 2005 04:58:59 -0600 reply

Status: open => closed




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