caught a bug Integer(ii+nn^2) in MonsMats() should be Integer(ii+2**nn)
just happens to work out with nn=4 oops
On Thu, Jul 31, 2014 at 11:08 PM, Stephen Kauffman <strangerl...@gmail.com>
wrote:
> Thanks for the tips. I got rid of the execs. Made it more general to
> include non-othogonal generators and metric. Did some spot checks and seems
> to be working. Unfortunately for cliff_elt in the FreeAlgebraQuotient there
> is no cliff_elt.coefficients() or cliff_elt.coefficient(mon) nor
> cliff_elt.generators() and for cliff_vect*cliff_vect which is scalar
> ostensibly in the base ring but can't divide or coerce back into the base
> ring. Had to write special functions. Also for n generators there are n
> structure matrices which are (2^n) x (2^n) to define the quotient, and I
> don't know the practical limits on the number of generators
>
> #Clifford Algebra construction with FreeAlgebraQuotient both orthogonal or
> non-orthogonal
> def GenMul(free_gen1,free_gen2):
> ii = F.gens().index(free_gen1)
> jj = F.gens().index(free_gen2)
> if free_gen1==free_gen2:
> return Metric[ii,jj]
> else:
> return MGGG[ii,jj]
>
> def GenList(bit_str,Struct):
> gen_list = []
>
> for jj in range(nn):
> if bit_str[jj]=='0':
> gen_list.append(Struct(1))
> else:
> gen_list.append(Struct.gen(nn-jj-1))
> return gen_list[::-1]
>
>
> def MonProduct(mon_list):
> mon_prod = mon_list[0]
> for jj in range(1,len(mon_list)):
> mon_prod = mon_prod*mon_list[jj]
> return mon_prod
>
> def MonsMats():
> mons_str = []
> mons_list = []
> mons = []
> for ii in range(2**nn):
> bit_str = Integer(ii+nn^2).binary()[1:nn+1] # reversed
> '1011' corresponds to g0*g1*g3 etc
> mons_list.append(GenList(bit_str,PRGA))
> mons.append(MonProduct(GenList(bit_str,F)))
> mons_str = [str(mons[kk]) for kk in range(len(mons))]
> mons_str_sorted = sorted(sorted(mons_str),key=len)
> # sort mons lexicographically then by lowest degree first
> mons_list = [mons_list[[mons_str.index(mons_str_sorted[kk]) for kk in
> range(len(mons_str))][ll]] for ll in range(len(mons_str))]
>
> mons = [mons[[mons_str.index(mons_str_sorted[kk]) for kk in
> range(len(mons_str))][ll]] for ll in range(len(mons_str))]
> mats = []
> for g in PRGA.gens():
> xgen = []
> for m in mons_list:
> m2 = m[:] # make copy by value not ref
> res1 = []
> for ii in range(len(m)-1, PRGA.gens().index(g)-1,-1):
> if g < m[ii]:
> res1.append((GenMul(m2[ii],g) +
> g*m2[ii])*MonProduct(m2[0:ii]+m2[ii+1:len(m2)]))
> m2[ii] = -m2[ii]
> elif m[ii]==1 and PRGA.gens().index(g) != ii:
> continue
> elif m[ii]==1 and PRGA.gens().index(g) == ii:
> m2[ii] = g
> break
> else:
> m2[ii] = GenMul(g,g)
> break
> xgen.append(sum(res1) + MonProduct(m2))
> # nn (2^nn x 2^nn) structure matrices ? practical limit on number
> nn of generators ?
> mats.append(matrix([[mxg.coefficient(m) for m in mons] for mxg in
> xgen]))
> gen_names = tuple(str(g) for g in PRGA.gens())
> return [mons, mats, gen_names]
>
> def cliff_coef(cliff_elt):
> Struct=cliff_elt.parent()
> return tuple(Struct.free_algebra()(str(cliff_elt)).coefficient(mon)
> for mon in Struct.monomial_basis())
>
> def cliff_inner(cliff_vect):
> Struct=cliff_vect.parent()
> return
> Struct.free_algebra()(str(cliff_vect*cliff_vect)).coefficient(Struct.monoid()(1))
>
> # random example
> BaseRing = QQ
> signat = diagonal_matrix(BaseRing,[1,-1,-1,-1])
> nn = signat.nrows()
> rank = 0
> while rank<nn:
> Metric = random_matrix(BaseRing,nn)
> rank = Metric.rank()
> Metric = Metric.transpose()*signat*Metric # somewhat random
> non-orthogonal symmetric full rank Metric with signature defined above
> PRGA = FreeAlgebra(BaseRing,nn,'g')
> F = PRGA.monoid()
> F.inject_variables()
> MG = matrix(PRGA.gens())
> MGG = MG.transpose()*MG
> MGGG = 2*Metric - MGG.transpose() # global used in function
> GenMul(x,y) as is global Metric
> mons_mats = MonsMats()
> ST = FreeAlgebraQuotient(PRGA,mons_mats[0],mons_mats[1],names=mons_mats[2])
> ST.inject_variables()
>
> # examples
> cliff_elt = g3*g2*g1*g0
> [ST.free_algebra()(str(cliff_elt)).coefficient(mon) for mon in
> ST.monomial_basis()] # st_elt.coefficients() results in error since
> it has no such attribute
> # or
> tuple(ST.free_algebra()(str(cliff_elt)).coefficient(mon) for mon in
> ST.monomial_basis()) # is a possiblity
> print(cliff_elt);print(cliff_coef(cliff_elt))
>
> cliff_vect = 5*g0+7*g1-2*g2+6*g3/7
> # to get the inner product back in the base ring, can't divide say
> g3/(g3*g3) even though g3*g3 scalar ostensibly in base field but not...
> ST.free_algebra()(str(cliff_vect*cliff_vect)).coefficient(ST.monoid()(1))
>
> print(cliff_vect);print(cliff_coef(cliff_vect));print(cliff_inner(cliff_vect))
>
>
> On Tue, Jul 29, 2014 at 12:17 AM, Nils Bruin <nbr...@sfu.ca> wrote:
>
>> On Monday, July 28, 2014 7:24:47 PM UTC-7, Stephen Kauffman wrote:
>>>
>>>
>>> exec preparse('ST.<' + gen_str + '>=FreeAlgebraQuotient(PRGA,
>>> mons_mats[0],mons_mats[1])')
>>>
>>
>> Congratulations to get all of this figured out! It's nice to see code of
>> this generality.
>>
>> Please use
>>
>> gen_names=tuple(str(g) for g in PRGA.gens())
>> ST=FreeAlgebraQuotient(PRGA,mons_mats[0],mons_mats[1],names=gen_names)
>>
>> instead, and perhaps
>>
>> ST.inject_variables()
>>
>> if you want the generators bound to corresponding names in the current
>> global scope.
>>
>> You should really avoid "exec" whenever you can. For one thing, it makes
>> it really difficult for the Python compiler to do a decent job.
>>
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>
>
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