Mike,
sorry, i've been away from the internet, and also distracted,
or would have replied sooner. see below.
On Wed, 26 Dec 2007, Mike Hansen wrote:
>
> Hi Kyle,
>
> This is one area in which Sage (and GAP4 for that matter) could use
> some work -- there hasn't really been anyone been working on it.
> Right now, there is a FreeAlgebraQuotient (
> http://www.sagemath.org/doc/html/ref/module-sage.algebras.free-algebra-quotient.html
> ) in Sage which allows you to work with these objects assuming you
> have a faithful representation for the algebra. The vector
> enumeration algorithms allow one to go from a description of the
> underlying two sided ideal like you have with a*b-b*a-a to a
> representation that could be plugged into FreeAlgebraQuotient. There
> is a free software implementation in C by Steve Linton, and we should
> definitely think about including it in Sage. It comes with the GAP3
> package "ve".
this isn't my area at all, so i didn't really understand that, but doesn't
FreeAlgebraQuotient create only finite dimensional algebras, which mine is
not?
>
> The other way to go about doing things is with noncommutative Grobner
> bases which I would be pretty interested in. But, Sage does not have
> any support for these at the moment. The only real algorithm that I
> know of is due to Mora, but this is probably due to my relative
> unfamiliarity with the area. I do know that Bergman (
> http://servus.math.su.se/bergman/ ) and the GAP4 package GBNP (
> http://www.mathdox.org/products/gbnp/ ) can carry out these
> computations. I have no idea on their relative efficiency though.
i finally figured out how to construct the algebra in
singular/plural, and i can get the "gcd" by asking for a standard basis
for the ideal generated by a given set of elements. but it seems like
that's about all i can do in singular. not only that, but i want the
output grouped by powers of a, say like (b^2 + 1)*a^2 + (b+1)*a + 1, and i
can't see how to even do that.
>
> It may come as no surprise, but Magma is probably the best software
> now in this area. Reading Magma's docs on this stuff is a good place
> to start http://www.math.uiuc.edu/Software/magma/text433.html .
too bad it costs money. i do have access to it through the school, but i
want something i can use on my laptop.
>
> --Mike
>
> P.S. I'm not sure what you (specifically) want when you say factoring
> or gcd in these algebras.
yes, yes... good question... mainly i want a
factorization into factors with lower degree in a, if one exists. in
my example, a and b have infinite order, so i think this makes sense.
i will now peruse your next message about GAP and try to digest it.
thanks
-kyle
>
>
> On Dec 26, 2007 1:03 AM, Kyle Schalm <[EMAIL PROTECTED]> wrote:
>>
>>
>> what's the status of noncommutative algebra in sage?
>>
>> suppose i want to play with expressions in the free algebra Z<a,b> modulo
>> the relation ab=ba+a. i'd also like to factor and gcd, ideally. er... no
>> pun intended.
>>
>> how do i do this -- or if sage can't do this (yet), what are my
>> options?
>> thanks
>> kyle
>>
>>
>>
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