plenty to read in http://docs.sympy.org/latest/modules/galgebra/GA.html
had to dig to see what to import
import sympy
from sympy import *
from sympy.galgebra import *
from sympy.galgebra.ga import *
some references said to
from sympy.galgebra.GAsympy import *
which doesn't exist or at least doesn't exist now. I will play around with
it, thanks, but now I am curious about free algebra quotients a little.
On Sat, Jul 26, 2014 at 8:27 PM, David Joyner <wdjoy...@gmail.com> wrote:
> Have you looked at what sympy already has for Clifford algebras?
>
>
> On Saturday, July 26, 2014, Stephen Kauffman <strangerl...@gmail.com>
> wrote:
>
>> I attempted to create a Clifford Algebra for space-time with the gamma
>> matrices using the FreeAlgebraQuotient in analogy to the example for
>> constructing a quarternion algebra from the documentation with the code:
>>
>> PRGA=FreeAlgebra(QQ,4,'g')
>> F = PRGA.monoid()
>> g0,g1,g2,g3 = F.gens()
>> mons = [ F(1), g0, g1, g2, g3, g0*g1, g0*g2, g0*g3, g1*g2, g2*g3, g3*g1,
>> g0*g1*g2*g3*g0, g0*g1*g2*g3*g1, g0*g1*g2*g3*g2, g0*g1*g2*g3*g3, g0*g1*g2*g3]
>> G0=diagonal_matrix([1,1,1,1,-1,-1,-1,-1]) #8x8 gamma matrices
>> ZR=matrix(2,2,0)
>> EE=diagonal_matrix([1,1])
>> II=matrix([[0,1],[-1,0]])
>>
>> G1=block_matrix([[ZR,ZR,ZR,EE],[ZR,ZR,EE,ZR],[ZR,-EE,ZR,ZR],[-EE,ZR,ZR,ZR]],subdivide=False)
>>
>> G2=block_matrix([[ZR,ZR,ZR,-II],[ZR,ZR,II,ZR],[ZR,II,ZR,ZR],[-II,ZR,ZR,ZR]],subdivide=False)
>>
>> G3=block_matrix([[ZR,ZR,EE,ZR],[ZR,ZR,ZR,-EE],[-EE,ZR,ZR,ZR],[ZR,EE,ZR,ZR]],subdivide=False)
>> mats = [G0,-G1,-G2,-G3]
>> ST.<g0,g1,g2,g3> = FreeAlgebraQuotient(PRGA, mons, mats)
>> ST
>>
>> Free algebra quotient on 4 generators ('g0', 'g1', 'g2', 'g3') and
>> dimension 16 over Rational Field
>>
>> everything looked promising here but when I type
>>
>> g0*g1
>>
>> Traceback (click to the left of this block for traceback)
>> ...
>> TypeError: unsupported operand parent(s) for '*': 'Vector space of
>> dimension 16 over Rational Field' and 'Full MatrixSpace of 8 by 8 dense
>> matrices over Integer
>> Ring'
>>
>> I expected sage to parrot back
>>
>> g0*g1
>>
>> Seems like we ought to be able to use a Free Algebra and the code:
>>
>> PRGA=FreeAlgebra(QQ,4,'g')
>> MG=matrix(PRGA.gens())
>> MGG=MG.transpose()*MG
>> Metric = diagonal_matrix([1,-1,-1,-1])
>> MGGM=MGG+MGG.transpose()-2*Metric
>> [MGGM[i,j] for i in range(4) for j in range(4)]
>>
>> [-2 + 2*g0^2, g0*g1 + g1*g0, g0*g2 + g2*g0, g0*g3 + g3*g0, g0*g1 +
>> g1*g0, 2 + 2*g1^2, g1*g2 + g2*g1, g1*g3 + g3*g1, g0*g2 + g2*g0, g1*g2 +
>> g2*g1, 2 + 2*g2^2,
>> g2*g3 + g3*g2, g0*g3 + g3*g0, g1*g3 + g3*g1, g2*g3 + g3*g2, 2 + 2*g3^2]
>>
>> and use this like an ideal. Sage is Clifford averse.
>>
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