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SandBoxCliffordAlgebra

Edit detail for SandBoxCliffordAlgebra revision 4 of 9

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Editor: bfauser Time: 2009/11/03 08:16:59 GMT-8
Note:

added:
We want to test some properties of the CliffordAlgebra domain implemented in AXIOM(TM)


changed:
-1+1
DiagMat:=matrix[[1,0,0],[0,1,0],[0,0,1]]
DQ := quadraticForm DiagMat
CLDQ := CliffordAlgebra(3, Fraction(Integer), DQ)

changed:
-From page Fri Oct 30 19:39:56 -0700 2009
-From: page
-Date: Fri, 30 Oct 2009 19:39:56 -0700
-Subject: good test
-Message-ID: <20091030193956-0700@axiom-wiki.newsynthesis.org>
-
-It seems that this test is good. I get the same result!
-
CLDQ contains now the Clifford algebra constructor for the Clifford algebra CL(R^3,Q)
where Q is the diagonal quadratic form in the basis of generators

changed:
-1+1
basGen: List CLDQ :=[1, e(1), e(2), e(1)*e(2), e(3), e(1)*e(3), e(2)*e(3), e(1)*e(2)*e(3)]

added:

And we can now compute within this basis
\begin{axiom}
elem1: CLDQ :=2*e(1)+e(2)
elem2: CLDQ :=1/2+e(2)+3*e(1)*e(3)
elem1*elem2
\end{axiom}
which is fine.

Lets now see what happens if we change the basis. We define new generators
\begin{axiom}
f1:CLDQ := e(1)-e(2)
f2:CLDQ := e(2)-e(3)
f3:CLDQ := e(1)+e(2)+e(3)
lstFGen: List CLDQ := [f1, f2, f3]
\end{axiom}
and check what the new defining relations are
\begin{axiom}
matrix [[y*x for x in lstFGen] for y in lstFGen]
\end{axiom}

However, let us do the same calculations with another, symmetric but not diagonal quadratic form
\begin{axiom}
OffDiagMat:=matrix[[0,0,1],[0,1,0],[1,0,0]]
ODQ := quadraticForm OffDiagMat
CLODQ := CliffordAlgebra(3, Fraction(Integer), ODQ)
basGenO: List CLODQ :=[1, e(1), e(2), e(1)*e(2), e(3), e(1)*e(3), e(2)*e(3), e(1)*e(2)*e(3)]
basO: List CLODQ :=[e(1), e(2), e(3)]
\end{axiom}
and let us check the multiplication table on the basis and in general:
\begin{axiom}
matrix [[y*x for x in basO] for y in basO]
matrix [[y*x for x in basGenO] for y in basGenO]
\end{axiom}
which is irritation, since the Clifford product is just 'unevaluated', hence the
basis elements are generated by Cliffrord products. We would liek to see the 
result in a Grassmann basis
\begin{axiom}
fbasis:List CLODQ :=[ 1, e(1), e(2), 1/2*(e(1)*e(2)-e(2)*e(1), e(3), 1/2*(e(1)*e(3)-e(3)*e(1), 1/2*(e(2)*e(3)-e(3)*e(2),
                     1/6(e(1)*e(2)*e(3) + e(2)*e(3)*e(1) + e(3)*e(1)*e(2) - e(1)*e(3)*e(2) - e(3)*e(2)*e(1) - e(2)*e(1)*e(3)) ]
\end{axiom}

We want to test some properties of the CliffordAlgebra? domain implemented in AXIOM(TM)

axiom

DiagMat:=matrix[[1,0,0],[0,1,0],[0,0,1]]

(1)

Type: Matrix(Integer)

axiom

DQ := quadraticForm DiagMat

(2)

Type: QuadraticForm?(3,Fraction(Integer))

axiom

CLDQ := CliffordAlgebra(3, Fraction(Integer), DQ)

(3)

Type: Domain

CLDQ contains now the Clifford algebra constructor for the Clifford algebra CL(R^3,Q) where Q is the diagonal quadratic form in the basis of generators

axiom

basGen: List CLDQ :=[1, e(1), e(2), e(1)*e(2), e(3), e(1)*e(3), e(2)*e(3),
e(1)*e(2)*e(3)]

(4)

Type: List(CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?))

And we can now compute within this basis

axiom

elem1: CLDQ :=2*e(1)+e(2)

(5)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

axiom

elem2: CLDQ :=1/2+e(2)+3*e(1)*e(3)

(6)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

axiom

elem1*elem2

(7)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

which is fine.

Lets now see what happens if we change the basis. We define new generators

axiom

f1:CLDQ := e(1)-e(2)

(8)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

axiom

f2:CLDQ := e(2)-e(3)

(9)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

axiom

f3:CLDQ := e(1)+e(2)+e(3)

(10)

Type: CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?)

axiom

lstFGen: List CLDQ := [f1, f2, f3]

(11)

Type: List(CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?))

and check what the new defining relations are

axiom

matrix [[y*x for x in lstFGen] for y in lstFGen]

(12)

Type: Matrix(CliffordAlgebra?(3,Fraction(Integer),[[1,0,0],[0,1,0]?,[0,0,1]]?))

However, let us do the same calculations with another, symmetric but not diagonal quadratic form

axiom

OffDiagMat:=matrix[[0,0,1],[0,1,0],[1,0,0]]

(13)

Type: Matrix(Integer)

axiom

ODQ := quadraticForm OffDiagMat

(14)

Type: QuadraticForm?(3,Fraction(Integer))

axiom

CLODQ := CliffordAlgebra(3, Fraction(Integer), ODQ)

(15)

Type: Domain

axiom

basGenO: List CLODQ :=[1, e(1), e(2), e(1)*e(2), e(3), e(1)*e(3),
e(2)*e(3), e(1)*e(2)*e(3)]

(16)

Type: List(CliffordAlgebra?(3,Fraction(Integer),[[0,0,1],[0,1,0]?,[1,0,0]]?))

axiom

basO: List CLODQ :=[e(1), e(2), e(3)]

(17)

Type: List(CliffordAlgebra?(3,Fraction(Integer),[[0,0,1],[0,1,0]?,[1,0,0]]?))

and let us check the multiplication table on the basis and in general:

axiom

matrix [[y*x for x in basO] for y in basO]

(18)

Type: Matrix(CliffordAlgebra?(3,Fraction(Integer),[[0,0,1],[0,1,0]?,[1,0,0]]?))

axiom

matrix [[y*x for x in basGenO] for y in basGenO]

(19)

Type: Matrix(CliffordAlgebra?(3,Fraction(Integer),[[0,0,1],[0,1,0]?,[1,0,0]]?))

which is irritation, since the Clifford product is just unevaluated, hence the basis elements are generated by Cliffrord products. We would liek to see the result in a Grassmann basis

axiom

fbasis:List CLODQ :=[ 1, e(1), e(2), 1/2*(e(1)*e(2)-e(2)*e(1), e(3),
1/2*(e(1)*e(3)-e(3)*e(1), 1/2*(e(2)*e(3)-e(3)*e(2),
                     1/6(e(1)*e(2)*e(3) + e(2)*e(3)*e(1) +
e(3)*e(1)*e(2) - e(1)*e(3)*e(2) - e(3)*e(2)*e(1) -
e(2)*e(1)*e(3)) ]

  Line   1: fbasis:List CLODQ :=[ 1, e(1), e(2),
1/2*(e(1)*e(2)-e(2)*e(1), e(3), 1/2*(e(1)*e(3)-e(3)*e(1),
1/2*(e(2)*e(3)-e(3)*e(2),
           ....................................................................................
...............A
  Error  A: Missing mate.
  Line   2:                      1/6(e(1)*e(2)*e(3) + e(2)*e(3)*e(1) +
e(3)*e(1)*e(2) - e(1)*e(3)*e(2) - e(3)*e(2)*e(1) -
e(2)*e(1)*e(3)) ]
           ....................................................................................
..........................................A
  Error  A: syntax error at top level
  Error  A: Possibly missing a )
   3 error(s) parsing