I did a bit of playing around with the GAP package GBNP. It is pretty nice, but the interface is a little clunky. I think it could easily provide a lot of functionality to Sage so I am going to spend some of my time at the AMS meeting working on this.
I made a very basic spkg at http://sage.math.washington.edu/home/mhansen/gbnp-0.9.5.spkg . Here is an example in the quotient algebra you mentioned: sage: gap.eval('LoadPackage("GBNP")') 'true' sage: gen = [ [[1,2],[2,1],[1]], [1,-1,-1] ] sage: relations = gap( [ gen ] ) sage: print gap.eval('PrintNPList(%s)'%relations.name()) ab - ba - a sage: GB = gap.SGrobner(relations) sage: print gap.eval('PrintNPList(%s)'%GB.name()) ba - ab + a sage: p = gap.StrongNormalFormNP( [[[1,2]],[1]], GB ); print gap.eval('PrintNP(%s)'%p.name()) ab sage: p = gap.StrongNormalFormNP( [[[2,1]],[1]], GB ); print gap.eval('PrintNP(%s)'%p.name()) ab - a sage: p = gap.StrongNormalFormNP( [[[2,1,1,2,1,2]],[1]], GB ); print gap.eval('PrintNP(%s)'%p.name()) a^3b^3 - 4a^3b^2 + 3a^3b Your quotient algebra is infinite dimensional whose dimension grows like n^2. sage: gap.FinCheck(gap.LTermsNP(GB), 2) false You can get a basis for the components of degree 0 through 4: sage: B = gap.BaseQA(GB, 2, 10) sage: print gap.eval('PrintNPList(%s)'%B.name()) 1 a b a^2 ab b^2 a^3 a^2b ab^2 b^3 a^4 a^3b a^2b^2 ab^3 b^4 Check out the examples for some more ideas of what you can do with this: http://www.mathdox.org/products/gbnp/chapA.html --Mike On Dec 26, 2007 2:57 AM, Mike Hansen <[EMAIL PROTECTED]> 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". > > 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. > > 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 . > > --Mike > > P.S. I'm not sure what you (specifically) want when you say factoring > or gcd in these algebras. > > > > 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 > > > > > > > > > > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
